{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0001","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":1,\"B\":-10,\"C\":1,\"a\":-2,\"b\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 3, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-2, 3, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102001,"parameters":{"A":1,"B":-10,"C":1,"a":-2,"b":3,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":1,\"B\":-10,\"C\":1,\"a\":-2,\"b\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0002","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":2,\"B\":-9,\"C\":2,\"a\":-3,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 4, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-3, 4, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102002,"parameters":{"A":2,"B":-9,"C":2,"a":-3,"b":4,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":2,\"B\":-9,\"C\":2,\"a\":-3,\"b\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0003","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":3,\"B\":-8,\"C\":5,\"a\":-4,\"b\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 5, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-4, 5, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102003,"parameters":{"A":3,"B":-8,"C":5,"a":-4,"b":5,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":3,\"B\":-8,\"C\":5,\"a\":-4,\"b\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0004","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":4,\"B\":-7,\"C\":10,\"a\":-5,\"b\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 6, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-5, 6, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102004,"parameters":{"A":4,"B":-7,"C":10,"a":-5,"b":6,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":4,\"B\":-7,\"C\":10,\"a\":-5,\"b\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0005","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":5,\"B\":-6,\"C\":17,\"a\":-6,\"b\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 7, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-6, 7, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102005,"parameters":{"A":5,"B":-6,"C":17,"a":-6,"b":7,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":5,\"B\":-6,\"C\":17,\"a\":-6,\"b\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0006","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":1,\"B\":-5,\"C\":26,\"a\":-7,\"b\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 8, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-7, 8, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102006,"parameters":{"A":1,"B":-5,"C":26,"a":-7,"b":8,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":1,\"B\":-5,\"C\":26,\"a\":-7,\"b\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0007","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":2,\"B\":-4,\"C\":37,\"a\":-8,\"b\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 9, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-8, 9, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102007,"parameters":{"A":2,"B":-4,"C":37,"a":-8,"b":9,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":2,\"B\":-4,\"C\":37,\"a\":-8,\"b\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0008","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":3,\"B\":-3,\"C\":50,\"a\":-9,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 10, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-9, 10, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102008,"parameters":{"A":3,"B":-3,"C":50,"a":-9,"b":10,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":3,\"B\":-3,\"C\":50,\"a\":-9,\"b\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0009","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":4,\"B\":-2,\"C\":65,\"a\":-10,\"b\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 11, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-10, 11, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102009,"parameters":{"A":4,"B":-2,"C":65,"a":-10,"b":11,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":4,\"B\":-2,\"C\":65,\"a\":-10,\"b\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0010","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":5,\"B\":-1,\"C\":82,\"a\":-11,\"b\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 12, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-11, 12, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102010,"parameters":{"A":5,"B":-1,"C":82,"a":-11,"b":12,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":5,\"B\":-1,\"C\":82,\"a\":-11,\"b\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0011","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":1,\"B\":0,\"C\":101,\"a\":-12,\"b\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 13, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-12, 13, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102011,"parameters":{"A":1,"B":0,"C":101,"a":-12,"b":13,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":1,\"B\":0,\"C\":101,\"a\":-12,\"b\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0012","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":2,\"B\":1,\"C\":122,\"a\":-13,\"b\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 14, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-13, 14, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102012,"parameters":{"A":2,"B":1,"C":122,"a":-13,"b":14,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":2,\"B\":1,\"C\":122,\"a\":-13,\"b\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0013","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":3,\"B\":2,\"C\":145,\"a\":-14,\"b\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14, 15, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-14, 15, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102013,"parameters":{"A":3,"B":2,"C":145,"a":-14,"b":15,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":3,\"B\":2,\"C\":145,\"a\":-14,\"b\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0014","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":4,\"B\":3,\"C\":170,\"a\":-15,\"b\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 16, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-15, 16, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102014,"parameters":{"A":4,"B":3,"C":170,"a":-15,"b":16,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":4,\"B\":3,\"C\":170,\"a\":-15,\"b\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0015","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":5,\"B\":4,\"C\":197,\"a\":-16,\"b\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16, 17, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-16, 17, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102015,"parameters":{"A":5,"B":4,"C":197,"a":-16,"b":17,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":5,\"B\":4,\"C\":197,\"a\":-16,\"b\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0016","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":1,\"B\":5,\"C\":226,\"a\":-17,\"b\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 18, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-17, 18, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102016,"parameters":{"A":1,"B":5,"C":226,"a":-17,"b":18,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":1,\"B\":5,\"C\":226,\"a\":-17,\"b\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0017","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":2,\"B\":6,\"C\":257,\"a\":-18,\"b\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18, 19, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-18, 19, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102017,"parameters":{"A":2,"B":6,"C":257,"a":-18,"b":19,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":2,\"B\":6,\"C\":257,\"a\":-18,\"b\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0018","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":3,\"B\":7,\"C\":290,\"a\":-19,\"b\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 20, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-19, 20, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102018,"parameters":{"A":3,"B":7,"C":290,"a":-19,"b":20,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":3,\"B\":7,\"C\":290,\"a\":-19,\"b\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0019","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":4,\"B\":8,\"C\":325,\"a\":-20,\"b\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20, 21, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-20, 21, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102019,"parameters":{"A":4,"B":8,"C":325,"a":-20,"b":21,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":4,\"B\":8,\"C\":325,\"a\":-20,\"b\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0020","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":5,\"B\":9,\"C\":362,\"a\":-21,\"b\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 22, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-21, 22, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102020,"parameters":{"A":5,"B":9,"C":362,"a":-21,"b":22,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":5,\"B\":9,\"C\":362,\"a\":-21,\"b\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0021","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":1,\"B\":10,\"C\":401,\"a\":-22,\"b\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22, 23, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-22, 23, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102021,"parameters":{"A":1,"B":10,"C":401,"a":-22,"b":23,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":1,\"B\":10,\"C\":401,\"a\":-22,\"b\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0022","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":2,\"B\":11,\"C\":442,\"a\":-23,\"b\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 24, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-23, 24, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102022,"parameters":{"A":2,"B":11,"C":442,"a":-23,"b":24,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":2,\"B\":11,\"C\":442,\"a\":-23,\"b\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0023","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":3,\"B\":12,\"C\":485,\"a\":-24,\"b\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24, 25, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-24, 25, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102023,"parameters":{"A":3,"B":12,"C":485,"a":-24,"b":25,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":3,\"B\":12,\"C\":485,\"a\":-24,\"b\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0024","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":4,\"B\":13,\"C\":530,\"a\":-25,\"b\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 26, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-25, 26, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102024,"parameters":{"A":4,"B":13,"C":530,"a":-25,"b":26,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":4,\"B\":13,\"C\":530,\"a\":-25,\"b\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0025","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-hypothesis-audit；資料={\"A\":5,\"B\":14,\"C\":577,\"a\":-26,\"b\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-26, 27, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)","explanation_zh":"Wave6第四層核對=(-26, 27, 是, 是, 是, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-hypothesis-audit","seed":202611102025,"parameters":{"A":5,"B":14,"C":577,"a":-26,"b":27,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-hypothesis-audit","semantic_template":"qf-mod41-mvt-hypothesis-audit/{\"A\":5,\"B\":14,\"C\":577,\"a\":-26,\"b\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0026","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":1,\"B\":-10,\"C\":1,\"a\":-2,\"b\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, -20, 5, -9, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(25, -20, 5, -9, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102026,"parameters":{"A":1,"B":-10,"C":1,"a":-2,"b":3,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":1,\"B\":-10,\"C\":1,\"a\":-2,\"b\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0027","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":2,\"B\":-9,\"C\":2,\"a\":-3,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, -2, 7, -7, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(47, -2, 7, -7, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102027,"parameters":{"A":2,"B":-9,"C":2,"a":-3,"b":4,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":2,\"B\":-9,\"C\":2,\"a\":-3,\"b\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0028","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":3,\"B\":-8,\"C\":5,\"a\":-4,\"b\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(85, 40, 9, -5, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(85, 40, 9, -5, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102028,"parameters":{"A":3,"B":-8,"C":5,"a":-4,"b":5,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":3,\"B\":-8,\"C\":5,\"a\":-4,\"b\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0029","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":4,\"B\":-7,\"C\":10,\"a\":-5,\"b\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(145, 112, 11, -3, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(145, 112, 11, -3, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102029,"parameters":{"A":4,"B":-7,"C":10,"a":-5,"b":6,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":4,\"B\":-7,\"C\":10,\"a\":-5,\"b\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0030","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":5,\"B\":-6,\"C\":17,\"a\":-6,\"b\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(233, 220, 13, -1, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(233, 220, 13, -1, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102030,"parameters":{"A":5,"B":-6,"C":17,"a":-6,"b":7,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":5,\"B\":-6,\"C\":17,\"a\":-6,\"b\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0031","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":1,\"B\":-5,\"C\":26,\"a\":-7,\"b\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(110, 50, 15, -4, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(110, 50, 15, -4, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102031,"parameters":{"A":1,"B":-5,"C":26,"a":-7,"b":8,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":1,\"B\":-5,\"C\":26,\"a\":-7,\"b\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0032","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":2,\"B\":-4,\"C\":37,\"a\":-8,\"b\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(197, 163, 17, -2, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(197, 163, 17, -2, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102032,"parameters":{"A":2,"B":-4,"C":37,"a":-8,"b":9,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":2,\"B\":-4,\"C\":37,\"a\":-8,\"b\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0033","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":3,\"B\":-3,\"C\":50,\"a\":-9,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(320, 320, 19, 0, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(320, 320, 19, 0, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102033,"parameters":{"A":3,"B":-3,"C":50,"a":-9,"b":10,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":3,\"B\":-3,\"C\":50,\"a\":-9,\"b\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0034","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":4,\"B\":-2,\"C\":65,\"a\":-10,\"b\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(485, 527, 21, 2, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(485, 527, 21, 2, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102034,"parameters":{"A":4,"B":-2,"C":65,"a":-10,"b":11,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":4,\"B\":-2,\"C\":65,\"a\":-10,\"b\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0035","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":5,\"B\":-1,\"C\":82,\"a\":-11,\"b\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(698, 790, 23, 4, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(698, 790, 23, 4, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102035,"parameters":{"A":5,"B":-1,"C":82,"a":-11,"b":12,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":5,\"B\":-1,\"C\":82,\"a\":-11,\"b\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0036","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":1,\"B\":0,\"C\":101,\"a\":-12,\"b\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(245, 270, 25, 1, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(245, 270, 25, 1, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102036,"parameters":{"A":1,"B":0,"C":101,"a":-12,"b":13,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":1,\"B\":0,\"C\":101,\"a\":-12,\"b\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0037","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":2,\"B\":1,\"C\":122,\"a\":-13,\"b\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(447, 528, 27, 3, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(447, 528, 27, 3, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102037,"parameters":{"A":2,"B":1,"C":122,"a":-13,"b":14,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":2,\"B\":1,\"C\":122,\"a\":-13,\"b\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0038","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":3,\"B\":2,\"C\":145,\"a\":-14,\"b\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(705, 850, 29, 5, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(705, 850, 29, 5, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102038,"parameters":{"A":3,"B":2,"C":145,"a":-14,"b":15,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":3,\"B\":2,\"C\":145,\"a\":-14,\"b\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0039","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":4,\"B\":3,\"C\":170,\"a\":-15,\"b\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1025, 1242, 31, 7, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(1025, 1242, 31, 7, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102039,"parameters":{"A":4,"B":3,"C":170,"a":-15,"b":16,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":4,\"B\":3,\"C\":170,\"a\":-15,\"b\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0040","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":5,\"B\":4,\"C\":197,\"a\":-16,\"b\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1413, 1710, 33, 9, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(1413, 1710, 33, 9, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102040,"parameters":{"A":5,"B":4,"C":197,"a":-16,"b":17,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":5,\"B\":4,\"C\":197,\"a\":-16,\"b\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0041","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":1,\"B\":5,\"C\":226,\"a\":-17,\"b\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(430, 640, 35, 6, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(430, 640, 35, 6, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102041,"parameters":{"A":1,"B":5,"C":226,"a":-17,"b":18,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":1,\"B\":5,\"C\":226,\"a\":-17,\"b\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0042","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":2,\"B\":6,\"C\":257,\"a\":-18,\"b\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(797, 1093, 37, 8, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(797, 1093, 37, 8, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102042,"parameters":{"A":2,"B":6,"C":257,"a":-18,"b":19,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":2,\"B\":6,\"C\":257,\"a\":-18,\"b\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0043","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":3,\"B\":7,\"C\":290,\"a\":-19,\"b\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1240, 1630, 39, 10, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(1240, 1630, 39, 10, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102043,"parameters":{"A":3,"B":7,"C":290,"a":-19,"b":20,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":3,\"B\":7,\"C\":290,\"a\":-19,\"b\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0044","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":4,\"B\":8,\"C\":325,\"a\":-20,\"b\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1765, 2257, 41, 12, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(1765, 2257, 41, 12, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102044,"parameters":{"A":4,"B":8,"C":325,"a":-20,"b":21,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":4,\"B\":8,\"C\":325,\"a\":-20,\"b\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0045","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":5,\"B\":9,\"C\":362,\"a\":-21,\"b\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2378, 2980, 43, 14, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(2378, 2980, 43, 14, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102045,"parameters":{"A":5,"B":9,"C":362,"a":-21,"b":22,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":5,\"B\":9,\"C\":362,\"a\":-21,\"b\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0046","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":1,\"B\":10,\"C\":401,\"a\":-22,\"b\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(665, 1160, 45, 11, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(665, 1160, 45, 11, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102046,"parameters":{"A":1,"B":10,"C":401,"a":-22,"b":23,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":1,\"B\":10,\"C\":401,\"a\":-22,\"b\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0047","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":2,\"B\":11,\"C\":442,\"a\":-23,\"b\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1247, 1858, 47, 13, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(1247, 1858, 47, 13, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102047,"parameters":{"A":2,"B":11,"C":442,"a":-23,"b":24,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":2,\"B\":11,\"C\":442,\"a\":-23,\"b\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0048","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":3,\"B\":12,\"C\":485,\"a\":-24,\"b\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1925, 2660, 49, 15, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(1925, 2660, 49, 15, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102048,"parameters":{"A":3,"B":12,"C":485,"a":-24,"b":25,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":3,\"B\":12,\"C\":485,\"a\":-24,\"b\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0049","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":4,\"B\":13,\"C\":530,\"a\":-25,\"b\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2705, 3572, 51, 17, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(2705, 3572, 51, 17, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102049,"parameters":{"A":4,"B":13,"C":530,"a":-25,"b":26,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":4,\"B\":13,\"C\":530,\"a\":-25,\"b\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0050","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-secant-slope；資料={\"A\":5,\"B\":14,\"C\":577,\"a\":-26,\"b\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3593, 4600, 53, 19, exact_endpoint_secant_slope)","explanation_zh":"Wave6第四層核對=(3593, 4600, 53, 19, exact_endpoint_secant_slope)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-secant-slope","seed":202611102050,"parameters":{"A":5,"B":14,"C":577,"a":-26,"b":27,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-secant-slope","semantic_template":"qf-mod41-mvt-secant-slope/{\"A\":5,\"B\":14,\"C\":577,\"a\":-26,\"b\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0051","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":1,\"B\":-10,\"C\":1,\"a\":-2,\"b\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, -9, -9, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, -9, -9, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102051,"parameters":{"A":1,"B":-10,"C":1,"a":-2,"b":3,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":1,\"B\":-10,\"C\":1,\"a\":-2,\"b\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0052","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":2,\"B\":-9,\"C\":2,\"a\":-3,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, -7, -7, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, -7, -7, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102052,"parameters":{"A":2,"B":-9,"C":2,"a":-3,"b":4,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":2,\"B\":-9,\"C\":2,\"a\":-3,\"b\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0053","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":3,\"B\":-8,\"C\":5,\"a\":-4,\"b\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, -5, -5, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, -5, -5, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102053,"parameters":{"A":3,"B":-8,"C":5,"a":-4,"b":5,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":3,\"B\":-8,\"C\":5,\"a\":-4,\"b\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0054","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":4,\"B\":-7,\"C\":10,\"a\":-5,\"b\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, -3, -3, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, -3, -3, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102054,"parameters":{"A":4,"B":-7,"C":10,"a":-5,"b":6,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":4,\"B\":-7,\"C\":10,\"a\":-5,\"b\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0055","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":5,\"B\":-6,\"C\":17,\"a\":-6,\"b\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, -1, -1, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, -1, -1, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102055,"parameters":{"A":5,"B":-6,"C":17,"a":-6,"b":7,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":5,\"B\":-6,\"C\":17,\"a\":-6,\"b\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0056","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":1,\"B\":-5,\"C\":26,\"a\":-7,\"b\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, -4, -4, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, -4, -4, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102056,"parameters":{"A":1,"B":-5,"C":26,"a":-7,"b":8,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":1,\"B\":-5,\"C\":26,\"a\":-7,\"b\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0057","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":2,\"B\":-4,\"C\":37,\"a\":-8,\"b\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, -2, -2, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, -2, -2, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102057,"parameters":{"A":2,"B":-4,"C":37,"a":-8,"b":9,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":2,\"B\":-4,\"C\":37,\"a\":-8,\"b\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0058","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":3,\"B\":-3,\"C\":50,\"a\":-9,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 0, 0, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 0, 0, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102058,"parameters":{"A":3,"B":-3,"C":50,"a":-9,"b":10,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":3,\"B\":-3,\"C\":50,\"a\":-9,\"b\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0059","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":4,\"B\":-2,\"C\":65,\"a\":-10,\"b\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 2, 2, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 2, 2, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102059,"parameters":{"A":4,"B":-2,"C":65,"a":-10,"b":11,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":4,\"B\":-2,\"C\":65,\"a\":-10,\"b\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0060","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":5,\"B\":-1,\"C\":82,\"a\":-11,\"b\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 4, 4, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 4, 4, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102060,"parameters":{"A":5,"B":-1,"C":82,"a":-11,"b":12,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":5,\"B\":-1,\"C\":82,\"a\":-11,\"b\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0061","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":1,\"B\":0,\"C\":101,\"a\":-12,\"b\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 1, 1, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 1, 1, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102061,"parameters":{"A":1,"B":0,"C":101,"a":-12,"b":13,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":1,\"B\":0,\"C\":101,\"a\":-12,\"b\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0062","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":2,\"B\":1,\"C\":122,\"a\":-13,\"b\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 3, 3, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 3, 3, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102062,"parameters":{"A":2,"B":1,"C":122,"a":-13,"b":14,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":2,\"B\":1,\"C\":122,\"a\":-13,\"b\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0063","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":3,\"B\":2,\"C\":145,\"a\":-14,\"b\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 5, 5, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 5, 5, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102063,"parameters":{"A":3,"B":2,"C":145,"a":-14,"b":15,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":3,\"B\":2,\"C\":145,\"a\":-14,\"b\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0064","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":4,\"B\":3,\"C\":170,\"a\":-15,\"b\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 7, 7, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 7, 7, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102064,"parameters":{"A":4,"B":3,"C":170,"a":-15,"b":16,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":4,\"B\":3,\"C\":170,\"a\":-15,\"b\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0065","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":5,\"B\":4,\"C\":197,\"a\":-16,\"b\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 9, 9, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 9, 9, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102065,"parameters":{"A":5,"B":4,"C":197,"a":-16,"b":17,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":5,\"B\":4,\"C\":197,\"a\":-16,\"b\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0066","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":1,\"B\":5,\"C\":226,\"a\":-17,\"b\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 6, 6, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 6, 6, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102066,"parameters":{"A":1,"B":5,"C":226,"a":-17,"b":18,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":1,\"B\":5,\"C\":226,\"a\":-17,\"b\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0067","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":2,\"B\":6,\"C\":257,\"a\":-18,\"b\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 8, 8, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 8, 8, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102067,"parameters":{"A":2,"B":6,"C":257,"a":-18,"b":19,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":2,\"B\":6,\"C\":257,\"a\":-18,\"b\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0068","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":3,\"B\":7,\"C\":290,\"a\":-19,\"b\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 10, 10, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 10, 10, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102068,"parameters":{"A":3,"B":7,"C":290,"a":-19,"b":20,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":3,\"B\":7,\"C\":290,\"a\":-19,\"b\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0069","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":4,\"B\":8,\"C\":325,\"a\":-20,\"b\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 12, 12, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 12, 12, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102069,"parameters":{"A":4,"B":8,"C":325,"a":-20,"b":21,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":4,\"B\":8,\"C\":325,\"a\":-20,\"b\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0070","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":5,\"B\":9,\"C\":362,\"a\":-21,\"b\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 14, 14, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 14, 14, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102070,"parameters":{"A":5,"B":9,"C":362,"a":-21,"b":22,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":5,\"B\":9,\"C\":362,\"a\":-21,\"b\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0073","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":3,\"B\":12,\"C\":485,\"a\":-24,\"b\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 15, 15, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 15, 15, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102073,"parameters":{"A":3,"B":12,"C":485,"a":-24,"b":25,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":3,\"B\":12,\"C\":485,\"a\":-24,\"b\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0074","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":4,\"B\":13,\"C\":530,\"a\":-25,\"b\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 17, 17, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 17, 17, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102074,"parameters":{"A":4,"B":13,"C":530,"a":-25,"b":26,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":4,\"B\":13,\"C\":530,\"a\":-25,\"b\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0075","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-polynomial-witness；資料={\"A\":5,\"B\":14,\"C\":577,\"a\":-26,\"b\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, 是, 19, 19, 是, quadratic_midpoint_witness)","explanation_zh":"Wave6第四層核對=(1/2, 是, 19, 19, 是, quadratic_midpoint_witness)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-polynomial-witness","seed":202611102075,"parameters":{"A":5,"B":14,"C":577,"a":-26,"b":27,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-polynomial-witness","semantic_template":"qf-mod41-mvt-polynomial-witness/{\"A\":5,\"B\":14,\"C\":577,\"a\":-26,\"b\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0076","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":0,\"b\":0,\"scale\":1,\"offset\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 0, 0, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 0, 0, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102076,"parameters":{"mode":"degenerate_interval","a":0,"b":0,"scale":1,"offset":0,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":0,\"b\":0,\"mode\":\"degenerate_interval\",\"offset\":0,\"scale\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0077","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"jump_at_zero\",\"a\":-2,\"b\":3,\"scale\":2,\"offset\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(jump_at_zero, -2, 3, 2, 1, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)","explanation_zh":"Wave6第四層核對=(jump_at_zero, -2, 3, 2, 1, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102077,"parameters":{"mode":"jump_at_zero","a":-2,"b":3,"scale":2,"offset":1,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-2,\"b\":3,\"mode\":\"jump_at_zero\",\"offset\":1,\"scale\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0078","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"cusp_at_zero\",\"a\":-3,\"b\":4,\"scale\":3,\"offset\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cusp_at_zero, -3, 4, 3, 2, 是, 否, 否, open_interval_differentiability_fails_at_zero)","explanation_zh":"Wave6第四層核對=(cusp_at_zero, -3, 4, 3, 2, 是, 否, 否, open_interval_differentiability_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102078,"parameters":{"mode":"cusp_at_zero","a":-3,"b":4,"scale":3,"offset":2,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-3,\"b\":4,\"mode\":\"cusp_at_zero\",\"offset\":2,\"scale\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0079","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":3,\"b\":3,\"scale\":4,\"offset\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 3, 3, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 3, 3, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102079,"parameters":{"mode":"degenerate_interval","a":3,"b":3,"scale":4,"offset":-3,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":3,\"b\":3,\"mode\":\"degenerate_interval\",\"offset\":-3,\"scale\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0080","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"jump_at_zero\",\"a\":-5,\"b\":6,\"scale\":5,\"offset\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(jump_at_zero, -5, 6, 5, 4, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)","explanation_zh":"Wave6第四層核對=(jump_at_zero, -5, 6, 5, 4, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102080,"parameters":{"mode":"jump_at_zero","a":-5,"b":6,"scale":5,"offset":4,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-5,\"b\":6,\"mode\":\"jump_at_zero\",\"offset\":4,\"scale\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0081","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"cusp_at_zero\",\"a\":-6,\"b\":7,\"scale\":6,\"offset\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cusp_at_zero, -6, 7, 6, 5, 是, 否, 否, open_interval_differentiability_fails_at_zero)","explanation_zh":"Wave6第四層核對=(cusp_at_zero, -6, 7, 6, 5, 是, 否, 否, open_interval_differentiability_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102081,"parameters":{"mode":"cusp_at_zero","a":-6,"b":7,"scale":6,"offset":5,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-6,\"b\":7,\"mode\":\"cusp_at_zero\",\"offset\":5,\"scale\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0082","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":6,\"b\":6,\"scale\":7,\"offset\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 6, 6, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 6, 6, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102082,"parameters":{"mode":"degenerate_interval","a":6,"b":6,"scale":7,"offset":-6,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":6,\"b\":6,\"mode\":\"degenerate_interval\",\"offset\":-6,\"scale\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0083","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"jump_at_zero\",\"a\":-8,\"b\":9,\"scale\":8,\"offset\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(jump_at_zero, -8, 9, 8, 7, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)","explanation_zh":"Wave6第四層核對=(jump_at_zero, -8, 9, 8, 7, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102083,"parameters":{"mode":"jump_at_zero","a":-8,"b":9,"scale":8,"offset":7,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-8,\"b\":9,\"mode\":\"jump_at_zero\",\"offset\":7,\"scale\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0084","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"cusp_at_zero\",\"a\":-9,\"b\":10,\"scale\":9,\"offset\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cusp_at_zero, -9, 10, 9, 8, 是, 否, 否, open_interval_differentiability_fails_at_zero)","explanation_zh":"Wave6第四層核對=(cusp_at_zero, -9, 10, 9, 8, 是, 否, 否, open_interval_differentiability_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102084,"parameters":{"mode":"cusp_at_zero","a":-9,"b":10,"scale":9,"offset":8,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-9,\"b\":10,\"mode\":\"cusp_at_zero\",\"offset\":8,\"scale\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0085","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":9,\"b\":9,\"scale\":10,\"offset\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 9, 9, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 9, 9, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102085,"parameters":{"mode":"degenerate_interval","a":9,"b":9,"scale":10,"offset":-9,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":9,\"b\":9,\"mode\":\"degenerate_interval\",\"offset\":-9,\"scale\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0086","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"jump_at_zero\",\"a\":-11,\"b\":12,\"scale\":11,\"offset\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(jump_at_zero, -11, 12, 11, 10, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)","explanation_zh":"Wave6第四層核對=(jump_at_zero, -11, 12, 11, 10, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102086,"parameters":{"mode":"jump_at_zero","a":-11,"b":12,"scale":11,"offset":10,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-11,\"b\":12,\"mode\":\"jump_at_zero\",\"offset\":10,\"scale\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0087","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"cusp_at_zero\",\"a\":-12,\"b\":13,\"scale\":12,\"offset\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cusp_at_zero, -12, 13, 12, 11, 是, 否, 否, open_interval_differentiability_fails_at_zero)","explanation_zh":"Wave6第四層核對=(cusp_at_zero, -12, 13, 12, 11, 是, 否, 否, open_interval_differentiability_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102087,"parameters":{"mode":"cusp_at_zero","a":-12,"b":13,"scale":12,"offset":11,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-12,\"b\":13,\"mode\":\"cusp_at_zero\",\"offset\":11,\"scale\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0088","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":12,\"b\":12,\"scale\":13,\"offset\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 12, 12, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 12, 12, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102088,"parameters":{"mode":"degenerate_interval","a":12,"b":12,"scale":13,"offset":-12,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":12,\"b\":12,\"mode\":\"degenerate_interval\",\"offset\":-12,\"scale\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0089","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"jump_at_zero\",\"a\":-14,\"b\":15,\"scale\":14,\"offset\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(jump_at_zero, -14, 15, 14, 13, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)","explanation_zh":"Wave6第四層核對=(jump_at_zero, -14, 15, 14, 13, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102089,"parameters":{"mode":"jump_at_zero","a":-14,"b":15,"scale":14,"offset":13,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-14,\"b\":15,\"mode\":\"jump_at_zero\",\"offset\":13,\"scale\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0090","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"cusp_at_zero\",\"a\":-15,\"b\":16,\"scale\":15,\"offset\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cusp_at_zero, -15, 16, 15, 14, 是, 否, 否, open_interval_differentiability_fails_at_zero)","explanation_zh":"Wave6第四層核對=(cusp_at_zero, -15, 16, 15, 14, 是, 否, 否, open_interval_differentiability_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102090,"parameters":{"mode":"cusp_at_zero","a":-15,"b":16,"scale":15,"offset":14,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-15,\"b\":16,\"mode\":\"cusp_at_zero\",\"offset\":14,\"scale\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0091","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":15,\"b\":15,\"scale\":16,\"offset\":-15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 15, 15, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 15, 15, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102091,"parameters":{"mode":"degenerate_interval","a":15,"b":15,"scale":16,"offset":-15,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":15,\"b\":15,\"mode\":\"degenerate_interval\",\"offset\":-15,\"scale\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0092","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"jump_at_zero\",\"a\":-17,\"b\":18,\"scale\":17,\"offset\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(jump_at_zero, -17, 18, 17, 16, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)","explanation_zh":"Wave6第四層核對=(jump_at_zero, -17, 18, 17, 16, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102092,"parameters":{"mode":"jump_at_zero","a":-17,"b":18,"scale":17,"offset":16,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-17,\"b\":18,\"mode\":\"jump_at_zero\",\"offset\":16,\"scale\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0093","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"cusp_at_zero\",\"a\":-18,\"b\":19,\"scale\":18,\"offset\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cusp_at_zero, -18, 19, 18, 17, 是, 否, 否, open_interval_differentiability_fails_at_zero)","explanation_zh":"Wave6第四層核對=(cusp_at_zero, -18, 19, 18, 17, 是, 否, 否, open_interval_differentiability_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102093,"parameters":{"mode":"cusp_at_zero","a":-18,"b":19,"scale":18,"offset":17,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-18,\"b\":19,\"mode\":\"cusp_at_zero\",\"offset\":17,\"scale\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0094","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":18,\"b\":18,\"scale\":19,\"offset\":-18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 18, 18, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 18, 18, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102094,"parameters":{"mode":"degenerate_interval","a":18,"b":18,"scale":19,"offset":-18,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":18,\"b\":18,\"mode\":\"degenerate_interval\",\"offset\":-18,\"scale\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0095","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"jump_at_zero\",\"a\":-20,\"b\":21,\"scale\":20,\"offset\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(jump_at_zero, -20, 21, 20, 19, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)","explanation_zh":"Wave6第四層核對=(jump_at_zero, -20, 21, 20, 19, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102095,"parameters":{"mode":"jump_at_zero","a":-20,"b":21,"scale":20,"offset":19,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-20,\"b\":21,\"mode\":\"jump_at_zero\",\"offset\":19,\"scale\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0096","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"cusp_at_zero\",\"a\":-21,\"b\":22,\"scale\":21,\"offset\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cusp_at_zero, -21, 22, 21, 20, 是, 否, 否, open_interval_differentiability_fails_at_zero)","explanation_zh":"Wave6第四層核對=(cusp_at_zero, -21, 22, 21, 20, 是, 否, 否, open_interval_differentiability_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102096,"parameters":{"mode":"cusp_at_zero","a":-21,"b":22,"scale":21,"offset":20,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-21,\"b\":22,\"mode\":\"cusp_at_zero\",\"offset\":20,\"scale\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0097","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":21,\"b\":21,\"scale\":22,\"offset\":-21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 21, 21, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 21, 21, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102097,"parameters":{"mode":"degenerate_interval","a":21,"b":21,"scale":22,"offset":-21,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":21,\"b\":21,\"mode\":\"degenerate_interval\",\"offset\":-21,\"scale\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0098","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"jump_at_zero\",\"a\":-23,\"b\":24,\"scale\":23,\"offset\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(jump_at_zero, -23, 24, 23, 22, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)","explanation_zh":"Wave6第四層核對=(jump_at_zero, -23, 24, 23, 22, 否, 否, 否, endpoint_interval_continuity_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102098,"parameters":{"mode":"jump_at_zero","a":-23,"b":24,"scale":23,"offset":22,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-23,\"b\":24,\"mode\":\"jump_at_zero\",\"offset\":22,\"scale\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0099","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"cusp_at_zero\",\"a\":-24,\"b\":25,\"scale\":24,\"offset\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cusp_at_zero, -24, 25, 24, 23, 是, 否, 否, open_interval_differentiability_fails_at_zero)","explanation_zh":"Wave6第四層核對=(cusp_at_zero, -24, 25, 24, 23, 是, 否, 否, open_interval_differentiability_fails_at_zero)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102099,"parameters":{"mode":"cusp_at_zero","a":-24,"b":25,"scale":24,"offset":23,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":-24,\"b\":25,\"mode\":\"cusp_at_zero\",\"offset\":23,\"scale\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r02-mean-value-theorem-0100","batch_id":"batch-20261014-advance-20261110-r02-mean-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-mean-value-theorem"],"title_zh":"微分中值定理","stem_zh":"f(t)=A*t^2+B*t+C、a<b、closed interval continuity與open interval differentiability。操作=mvt-failure-boundary；資料={\"mode\":\"degenerate_interval\",\"a\":24,\"b\":24,\"scale\":25,\"offset\":-24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(degenerate_interval, 24, 24, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)","explanation_zh":"Wave6第四層核對=(degenerate_interval, 24, 24, 否, null, 否, a_less_than_b_and_nonzero_secant_denominator_required)；依完整明示資料獨立重算；witness c必須嚴格位於(a,b)且導數等於exact secant slope；witness不必唯一；有限samples、a=b、jump或interior cusp不可套定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["微分中值定理"]},"variant":{"family_id":"qf-mod41-mvt-failure-boundary","seed":202611102100,"parameters":{"mode":"degenerate_interval","a":24,"b":24,"scale":25,"offset":-24,"domain":"Exact_polynomial_mean_value_theorem","operation":"mvt-failure-boundary","semantic_template":"qf-mod41-mvt-failure-boundary/{\"a\":24,\"b\":24,\"mode\":\"degenerate_interval\",\"offset\":-24,\"scale\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}