{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0001","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-1,0,1],\"x\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-1, 0, 1], 2, 3, 4, updated, 5/4, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-1, 0, 1], 2, 3, 4, updated, 5/4, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191001,"parameters":{"coefficients":[-1,0,1],"x":2,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-1,1],[0,1],[1,1]],\"x\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0002","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-2,0,1],\"x\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-2, 0, 1], 3, 7, 6, updated, 11/6, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-2, 0, 1], 3, 7, 6, updated, 11/6, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191002,"parameters":{"coefficients":[-2,0,1],"x":3,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-2,1],[0,1],[1,1]],\"x\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0003","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-3,0,1],\"x\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-3, 0, 1], 4, 13, 8, updated, 19/8, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-3, 0, 1], 4, 13, 8, updated, 19/8, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191003,"parameters":{"coefficients":[-3,0,1],"x":4,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-3,1],[0,1],[1,1]],\"x\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0004","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-4,0,1],\"x\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-4, 0, 1], 5, 21, 10, updated, 29/10, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-4, 0, 1], 5, 21, 10, updated, 29/10, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191004,"parameters":{"coefficients":[-4,0,1],"x":5,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-4,1],[0,1],[1,1]],\"x\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0005","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-5,0,1],\"x\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-5, 0, 1], 1, -4, 2, updated, 3, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-5, 0, 1], 1, -4, 2, updated, 3, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191005,"parameters":{"coefficients":[-5,0,1],"x":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-5,1],[0,1],[1,1]],\"x\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0006","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-6,0,1],\"x\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-6, 0, 1], 2, -2, 4, updated, 5/2, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-6, 0, 1], 2, -2, 4, updated, 5/2, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191006,"parameters":{"coefficients":[-6,0,1],"x":2,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-6,1],[0,1],[1,1]],\"x\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0007","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-7,0,1],\"x\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-7, 0, 1], 3, 2, 6, updated, 8/3, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-7, 0, 1], 3, 2, 6, updated, 8/3, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191007,"parameters":{"coefficients":[-7,0,1],"x":3,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-7,1],[0,1],[1,1]],\"x\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0008","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-8,0,1],\"x\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-8, 0, 1], 4, 8, 8, updated, 3, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-8, 0, 1], 4, 8, 8, updated, 3, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191008,"parameters":{"coefficients":[-8,0,1],"x":4,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-8,1],[0,1],[1,1]],\"x\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0009","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-9,0,1],\"x\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-9, 0, 1], 5, 16, 10, updated, 17/5, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-9, 0, 1], 5, 16, 10, updated, 17/5, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191009,"parameters":{"coefficients":[-9,0,1],"x":5,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-9,1],[0,1],[1,1]],\"x\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0010","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-10,0,1],\"x\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-10, 0, 1], 1, -9, 2, updated, 11/2, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-10, 0, 1], 1, -9, 2, updated, 11/2, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191010,"parameters":{"coefficients":[-10,0,1],"x":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-10,1],[0,1],[1,1]],\"x\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0011","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-11,0,1],\"x\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-11, 0, 1], 2, -7, 4, updated, 15/4, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-11, 0, 1], 2, -7, 4, updated, 15/4, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191011,"parameters":{"coefficients":[-11,0,1],"x":2,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-11,1],[0,1],[1,1]],\"x\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0012","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-12,0,1],\"x\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-12, 0, 1], 3, -3, 6, updated, 7/2, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-12, 0, 1], 3, -3, 6, updated, 7/2, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191012,"parameters":{"coefficients":[-12,0,1],"x":3,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-12,1],[0,1],[1,1]],\"x\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0013","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-13,0,1],\"x\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-13, 0, 1], 4, 3, 8, updated, 29/8, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-13, 0, 1], 4, 3, 8, updated, 29/8, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191013,"parameters":{"coefficients":[-13,0,1],"x":4,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-13,1],[0,1],[1,1]],\"x\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0014","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-14,0,1],\"x\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-14, 0, 1], 5, 11, 10, updated, 39/10, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-14, 0, 1], 5, 11, 10, updated, 39/10, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191014,"parameters":{"coefficients":[-14,0,1],"x":5,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-14,1],[0,1],[1,1]],\"x\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0015","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-15,0,1],\"x\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-15, 0, 1], 1, -14, 2, updated, 8, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-15, 0, 1], 1, -14, 2, updated, 8, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191015,"parameters":{"coefficients":[-15,0,1],"x":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-15,1],[0,1],[1,1]],\"x\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0016","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-16,0,1],\"x\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-16, 0, 1], 2, -12, 4, updated, 5, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-16, 0, 1], 2, -12, 4, updated, 5, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191016,"parameters":{"coefficients":[-16,0,1],"x":2,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-16,1],[0,1],[1,1]],\"x\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0017","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-17,0,1],\"x\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-17, 0, 1], 3, -8, 6, updated, 13/3, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-17, 0, 1], 3, -8, 6, updated, 13/3, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191017,"parameters":{"coefficients":[-17,0,1],"x":3,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-17,1],[0,1],[1,1]],\"x\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0018","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-18,0,1],\"x\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-18, 0, 1], 4, -2, 8, updated, 17/4, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-18, 0, 1], 4, -2, 8, updated, 17/4, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191018,"parameters":{"coefficients":[-18,0,1],"x":4,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-18,1],[0,1],[1,1]],\"x\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0019","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-19,0,1],\"x\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-19, 0, 1], 5, 6, 10, updated, 22/5, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-19, 0, 1], 5, 6, 10, updated, 22/5, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191019,"parameters":{"coefficients":[-19,0,1],"x":5,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-19,1],[0,1],[1,1]],\"x\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0020","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-20,0,1],\"x\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-20, 0, 1], 1, -19, 2, updated, 21/2, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-20, 0, 1], 1, -19, 2, updated, 21/2, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191020,"parameters":{"coefficients":[-20,0,1],"x":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-20,1],[0,1],[1,1]],\"x\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0021","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-21,0,1],\"x\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-21, 0, 1], 2, -17, 4, updated, 25/4, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-21, 0, 1], 2, -17, 4, updated, 25/4, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191021,"parameters":{"coefficients":[-21,0,1],"x":2,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-21,1],[0,1],[1,1]],\"x\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0022","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-22,0,1],\"x\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-22, 0, 1], 3, -13, 6, updated, 31/6, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-22, 0, 1], 3, -13, 6, updated, 31/6, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191022,"parameters":{"coefficients":[-22,0,1],"x":3,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-22,1],[0,1],[1,1]],\"x\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0023","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-23,0,1],\"x\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-23, 0, 1], 4, -7, 8, updated, 39/8, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-23, 0, 1], 4, -7, 8, updated, 39/8, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191023,"parameters":{"coefficients":[-23,0,1],"x":4,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-23,1],[0,1],[1,1]],\"x\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0024","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-24,0,1],\"x\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-24, 0, 1], 5, 1, 10, updated, 49/10, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-24, 0, 1], 5, 1, 10, updated, 49/10, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191024,"parameters":{"coefficients":[-24,0,1],"x":5,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-24,1],[0,1],[1,1]],\"x\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0025","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-step；明示資料={\"coefficients\":[-25,0,1],\"x\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-25, 0, 1], 1, -24, 2, updated, 13, root_test_precedes_division)","explanation_zh":"Wave6第十三層核對=([-25, 0, 1], 1, -24, 2, updated, 13, root_test_precedes_division)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-step","seed":202611191025,"parameters":{"coefficients":[-25,0,1],"x":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-step","semantic_template":"newton-step|{\"coefficients\":[[-25,1],[0,1],[1,1]],\"x\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0026","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-1,0,1],\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-1, 0, 1], 1, [[0, 2, 3, 4, updated, 5/4]], step_budget_exhausted, 5/4, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-1, 0, 1], 1, [[0, 2, 3, 4, updated, 5/4]], step_budget_exhausted, 5/4, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191026,"parameters":{"coefficients":[-1,0,1],"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-1,1],[0,1],[1,1]],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0027","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-2,0,1],\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-2, 0, 1], 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 11/6, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-2, 0, 1], 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 11/6, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191027,"parameters":{"coefficients":[-2,0,1],"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-2,1],[0,1],[1,1]],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0028","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-3,0,1],\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-3, 0, 1], 1, [[0, 4, 13, 8, updated, 19/8]], step_budget_exhausted, 19/8, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-3, 0, 1], 1, [[0, 4, 13, 8, updated, 19/8]], step_budget_exhausted, 19/8, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191028,"parameters":{"coefficients":[-3,0,1],"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-3,1],[0,1],[1,1]],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0029","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-4,0,1],\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-4, 0, 1], 1, [[0, 5, 21, 10, updated, 29/10]], step_budget_exhausted, 29/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191029,"parameters":{"coefficients":[-4,0,1],"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-4,1],[0,1],[1,1]],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0030","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-5,0,1],\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-5, 0, 1], 1, [[0, 1, -4, 2, updated, 3]], step_budget_exhausted, 3, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-5, 0, 1], 1, [[0, 1, -4, 2, updated, 3]], step_budget_exhausted, 3, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191030,"parameters":{"coefficients":[-5,0,1],"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-5,1],[0,1],[1,1]],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0031","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-6,0,1],\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-6, 0, 1], 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, 5/2, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-6, 0, 1], 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, 5/2, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191031,"parameters":{"coefficients":[-6,0,1],"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-6,1],[0,1],[1,1]],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0032","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-7,0,1],\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-7, 0, 1], 1, [[0, 3, 2, 6, updated, 8/3]], step_budget_exhausted, 8/3, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-7, 0, 1], 1, [[0, 3, 2, 6, updated, 8/3]], step_budget_exhausted, 8/3, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191032,"parameters":{"coefficients":[-7,0,1],"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-7,1],[0,1],[1,1]],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0033","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-8,0,1],\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-8, 0, 1], 1, [[0, 4, 8, 8, updated, 3]], step_budget_exhausted, 3, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-8, 0, 1], 1, [[0, 4, 8, 8, updated, 3]], step_budget_exhausted, 3, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191033,"parameters":{"coefficients":[-8,0,1],"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-8,1],[0,1],[1,1]],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0034","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-9,0,1],\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-9, 0, 1], 1, [[0, 5, 16, 10, updated, 17/5]], step_budget_exhausted, 17/5, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-9, 0, 1], 1, [[0, 5, 16, 10, updated, 17/5]], step_budget_exhausted, 17/5, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191034,"parameters":{"coefficients":[-9,0,1],"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-9,1],[0,1],[1,1]],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0035","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-10,0,1],\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-10, 0, 1], 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, 11/2, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-10, 0, 1], 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, 11/2, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191035,"parameters":{"coefficients":[-10,0,1],"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-10,1],[0,1],[1,1]],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0036","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-11,0,1],\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-11, 0, 1], 1, [[0, 2, -7, 4, updated, 15/4]], step_budget_exhausted, 15/4, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-11, 0, 1], 1, [[0, 2, -7, 4, updated, 15/4]], step_budget_exhausted, 15/4, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191036,"parameters":{"coefficients":[-11,0,1],"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-11,1],[0,1],[1,1]],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0037","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-12,0,1],\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-12, 0, 1], 1, [[0, 3, -3, 6, updated, 7/2]], step_budget_exhausted, 7/2, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-12, 0, 1], 1, [[0, 3, -3, 6, updated, 7/2]], step_budget_exhausted, 7/2, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191037,"parameters":{"coefficients":[-12,0,1],"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-12,1],[0,1],[1,1]],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0038","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-13,0,1],\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-13, 0, 1], 1, [[0, 4, 3, 8, updated, 29/8]], step_budget_exhausted, 29/8, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-13, 0, 1], 1, [[0, 4, 3, 8, updated, 29/8]], step_budget_exhausted, 29/8, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191038,"parameters":{"coefficients":[-13,0,1],"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-13,1],[0,1],[1,1]],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0039","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-14,0,1],\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-14, 0, 1], 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, 39/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-14, 0, 1], 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, 39/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191039,"parameters":{"coefficients":[-14,0,1],"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-14,1],[0,1],[1,1]],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0040","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-15,0,1],\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-15, 0, 1], 1, [[0, 1, -14, 2, updated, 8]], step_budget_exhausted, 8, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-15, 0, 1], 1, [[0, 1, -14, 2, updated, 8]], step_budget_exhausted, 8, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191040,"parameters":{"coefficients":[-15,0,1],"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-15,1],[0,1],[1,1]],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0041","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-16,0,1],\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-16, 0, 1], 1, [[0, 2, -12, 4, updated, 5]], step_budget_exhausted, 5, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-16, 0, 1], 1, [[0, 2, -12, 4, updated, 5]], step_budget_exhausted, 5, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191041,"parameters":{"coefficients":[-16,0,1],"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-16,1],[0,1],[1,1]],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0042","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-17,0,1],\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-17, 0, 1], 1, [[0, 3, -8, 6, updated, 13/3]], step_budget_exhausted, 13/3, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-17, 0, 1], 1, [[0, 3, -8, 6, updated, 13/3]], step_budget_exhausted, 13/3, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191042,"parameters":{"coefficients":[-17,0,1],"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-17,1],[0,1],[1,1]],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0043","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-18,0,1],\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-18, 0, 1], 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, 17/4, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-18, 0, 1], 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, 17/4, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191043,"parameters":{"coefficients":[-18,0,1],"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-18,1],[0,1],[1,1]],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0044","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-19,0,1],\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-19, 0, 1], 1, [[0, 5, 6, 10, updated, 22/5]], step_budget_exhausted, 22/5, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-19, 0, 1], 1, [[0, 5, 6, 10, updated, 22/5]], step_budget_exhausted, 22/5, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191044,"parameters":{"coefficients":[-19,0,1],"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-19,1],[0,1],[1,1]],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0045","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-20,0,1],\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-20, 0, 1], 1, [[0, 1, -19, 2, updated, 21/2]], step_budget_exhausted, 21/2, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-20, 0, 1], 1, [[0, 1, -19, 2, updated, 21/2]], step_budget_exhausted, 21/2, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191045,"parameters":{"coefficients":[-20,0,1],"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-20,1],[0,1],[1,1]],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0046","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-21,0,1],\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-21, 0, 1], 1, [[0, 2, -17, 4, updated, 25/4]], step_budget_exhausted, 25/4, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-21, 0, 1], 1, [[0, 2, -17, 4, updated, 25/4]], step_budget_exhausted, 25/4, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191046,"parameters":{"coefficients":[-21,0,1],"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-21,1],[0,1],[1,1]],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0047","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-22,0,1],\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-22, 0, 1], 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, 31/6, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-22, 0, 1], 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, 31/6, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191047,"parameters":{"coefficients":[-22,0,1],"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-22,1],[0,1],[1,1]],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0048","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-23,0,1],\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-23, 0, 1], 1, [[0, 4, -7, 8, updated, 39/8]], step_budget_exhausted, 39/8, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-23, 0, 1], 1, [[0, 4, -7, 8, updated, 39/8]], step_budget_exhausted, 39/8, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191048,"parameters":{"coefficients":[-23,0,1],"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-23,1],[0,1],[1,1]],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0049","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-24,0,1],\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-24, 0, 1], 1, [[0, 5, 1, 10, updated, 49/10]], step_budget_exhausted, 49/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-24, 0, 1], 1, [[0, 5, 1, 10, updated, 49/10]], step_budget_exhausted, 49/10, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191049,"parameters":{"coefficients":[-24,0,1],"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-24,1],[0,1],[1,1]],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0050","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-finite-orbit；明示資料={\"coefficients\":[-25,0,1],\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-25, 0, 1], 1, [[0, 1, -24, 2, updated, 13]], step_budget_exhausted, 13, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)","explanation_zh":"Wave6第十三層核對=([-25, 0, 1], 1, [[0, 1, -24, 2, updated, 13]], step_budget_exhausted, 13, 否, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-finite-orbit","seed":202611191050,"parameters":{"coefficients":[-25,0,1],"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-finite-orbit","semantic_template":"newton-finite-orbit|{\"coefficients\":[[-25,1],[0,1],[1,1]],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0051","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":2,\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 2, 1, [[0, 2, 2, updated, 3/2]], step_budget_exhausted, 3/2, 1/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(2, 2, 1, [[0, 2, 2, updated, 3/2]], step_budget_exhausted, 3/2, 1/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191051,"parameters":{"radicand":2,"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[2,1],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0052","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":3,\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 3, 1, [[0, 3, 6, updated, 2]], step_budget_exhausted, 2, 1, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(3, 3, 1, [[0, 3, 6, updated, 2]], step_budget_exhausted, 2, 1, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191052,"parameters":{"radicand":3,"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[3,1],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0053","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":4,\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(4, 4, 1, [[0, 4, 12, updated, 5/2]], step_budget_exhausted, 5/2, 9/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(4, 4, 1, [[0, 4, 12, updated, 5/2]], step_budget_exhausted, 5/2, 9/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191053,"parameters":{"radicand":4,"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[4,1],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0054","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":5,\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(5, 5, 1, [[0, 5, 20, updated, 3]], step_budget_exhausted, 3, 4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(5, 5, 1, [[0, 5, 20, updated, 3]], step_budget_exhausted, 3, 4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191054,"parameters":{"radicand":5,"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[5,1],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0055","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":6,\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(6, 1, 1, [[0, 1, -5, updated, 7/2]], step_budget_exhausted, 7/2, 25/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(6, 1, 1, [[0, 1, -5, updated, 7/2]], step_budget_exhausted, 7/2, 25/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191055,"parameters":{"radicand":6,"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[6,1],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0056","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":7,\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(7, 2, 1, [[0, 2, -3, updated, 11/4]], step_budget_exhausted, 11/4, 9/16, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(7, 2, 1, [[0, 2, -3, updated, 11/4]], step_budget_exhausted, 11/4, 9/16, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191056,"parameters":{"radicand":7,"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[7,1],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0057","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":8,\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(8, 3, 1, [[0, 3, 1, updated, 17/6]], step_budget_exhausted, 17/6, 1/36, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(8, 3, 1, [[0, 3, 1, updated, 17/6]], step_budget_exhausted, 17/6, 1/36, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191057,"parameters":{"radicand":8,"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[8,1],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0058","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":9,\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(9, 4, 1, [[0, 4, 7, updated, 25/8]], step_budget_exhausted, 25/8, 49/64, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(9, 4, 1, [[0, 4, 7, updated, 25/8]], step_budget_exhausted, 25/8, 49/64, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191058,"parameters":{"radicand":9,"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[9,1],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0059","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":10,\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(10, 5, 1, [[0, 5, 15, updated, 7/2]], step_budget_exhausted, 7/2, 9/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(10, 5, 1, [[0, 5, 15, updated, 7/2]], step_budget_exhausted, 7/2, 9/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191059,"parameters":{"radicand":10,"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[10,1],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0060","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":11,\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(11, 1, 1, [[0, 1, -10, updated, 6]], step_budget_exhausted, 6, 25, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(11, 1, 1, [[0, 1, -10, updated, 6]], step_budget_exhausted, 6, 25, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191060,"parameters":{"radicand":11,"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[11,1],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0061","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":12,\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(12, 2, 1, [[0, 2, -8, updated, 4]], step_budget_exhausted, 4, 4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(12, 2, 1, [[0, 2, -8, updated, 4]], step_budget_exhausted, 4, 4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191061,"parameters":{"radicand":12,"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[12,1],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0062","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":13,\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(13, 3, 1, [[0, 3, -4, updated, 11/3]], step_budget_exhausted, 11/3, 4/9, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(13, 3, 1, [[0, 3, -4, updated, 11/3]], step_budget_exhausted, 11/3, 4/9, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191062,"parameters":{"radicand":13,"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[13,1],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0063","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":14,\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(14, 4, 1, [[0, 4, 2, updated, 15/4]], step_budget_exhausted, 15/4, 1/16, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(14, 4, 1, [[0, 4, 2, updated, 15/4]], step_budget_exhausted, 15/4, 1/16, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191063,"parameters":{"radicand":14,"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[14,1],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0064","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":15,\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(15, 5, 1, [[0, 5, 10, updated, 4]], step_budget_exhausted, 4, 1, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(15, 5, 1, [[0, 5, 10, updated, 4]], step_budget_exhausted, 4, 1, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191064,"parameters":{"radicand":15,"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[15,1],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0065","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":16,\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(16, 1, 1, [[0, 1, -15, updated, 17/2]], step_budget_exhausted, 17/2, 225/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(16, 1, 1, [[0, 1, -15, updated, 17/2]], step_budget_exhausted, 17/2, 225/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191065,"parameters":{"radicand":16,"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[16,1],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0066","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":17,\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(17, 2, 1, [[0, 2, -13, updated, 21/4]], step_budget_exhausted, 21/4, 169/16, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(17, 2, 1, [[0, 2, -13, updated, 21/4]], step_budget_exhausted, 21/4, 169/16, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191066,"parameters":{"radicand":17,"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[17,1],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0067","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":18,\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(18, 3, 1, [[0, 3, -9, updated, 9/2]], step_budget_exhausted, 9/2, 9/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(18, 3, 1, [[0, 3, -9, updated, 9/2]], step_budget_exhausted, 9/2, 9/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191067,"parameters":{"radicand":18,"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[18,1],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0068","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":19,\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(19, 4, 1, [[0, 4, -3, updated, 35/8]], step_budget_exhausted, 35/8, 9/64, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(19, 4, 1, [[0, 4, -3, updated, 35/8]], step_budget_exhausted, 35/8, 9/64, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191068,"parameters":{"radicand":19,"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[19,1],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0069","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":20,\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191069,"parameters":{"radicand":20,"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[20,1],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0070","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":21,\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(21, 1, 1, [[0, 1, -20, updated, 11]], step_budget_exhausted, 11, 100, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(21, 1, 1, [[0, 1, -20, updated, 11]], step_budget_exhausted, 11, 100, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191070,"parameters":{"radicand":21,"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[21,1],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0071","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":22,\"steps\":1,\"x0\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(22, 2, 1, [[0, 2, -18, updated, 13/2]], step_budget_exhausted, 13/2, 81/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(22, 2, 1, [[0, 2, -18, updated, 13/2]], step_budget_exhausted, 13/2, 81/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191071,"parameters":{"radicand":22,"x0":2,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[22,1],\"steps\":1,\"x0\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0072","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":23,\"steps\":1,\"x0\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(23, 3, 1, [[0, 3, -14, updated, 16/3]], step_budget_exhausted, 16/3, 49/9, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(23, 3, 1, [[0, 3, -14, updated, 16/3]], step_budget_exhausted, 16/3, 49/9, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191072,"parameters":{"radicand":23,"x0":3,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[23,1],\"steps\":1,\"x0\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0073","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":24,\"steps\":1,\"x0\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(24, 4, 1, [[0, 4, -8, updated, 5]], step_budget_exhausted, 5, 1, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(24, 4, 1, [[0, 4, -8, updated, 5]], step_budget_exhausted, 5, 1, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191073,"parameters":{"radicand":24,"x0":4,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[24,1],\"steps\":1,\"x0\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0074","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":25,\"steps\":1,\"x0\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(25, 5, 1, [[0, 5, 0, root_found, null]], root_found, 5, 0, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(25, 5, 1, [[0, 5, 0, root_found, null]], root_found, 5, 0, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191074,"parameters":{"radicand":25,"x0":5,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[25,1],\"steps\":1,\"x0\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0075","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-square-root-orbit；明示資料={\"radicand\":26,\"steps\":1,\"x0\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(26, 1, 1, [[0, 1, -25, updated, 27/2]], step_budget_exhausted, 27/2, 625/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)","explanation_zh":"Wave6第十三層核對=(26, 1, 1, [[0, 1, -25, updated, 27/2]], step_budget_exhausted, 27/2, 625/4, 否, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-square-root-orbit","seed":202611191075,"parameters":{"radicand":26,"x0":1,"steps":1,"domain":"Exact_Newton_iteration_boundary","operation":"newton-square-root-orbit","semantic_template":"newton-square-root-orbit|{\"radicand\":[26,1],\"steps\":1,\"x0\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0076","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[1,0,1],\"x\":0}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_derivative, 0, 1, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)","explanation_zh":"Wave6第十三層核對=(zero_derivative, 0, 1, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191076,"parameters":{"mode":"zero_derivative","model":{"coefficients":[1,0,1],"x":0},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[[1,1],[0,1],[1,1]],\"x\":[0,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0077","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[-2,0,1],\"steps\":1,\"x0\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191077,"parameters":{"mode":"finite_trace_only","model":{"coefficients":[-2,0,1],"x0":3,"steps":1},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[[-2,1],[0,1],[1,1]],\"steps\":1,\"x0\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0078","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"root_stop\",\"model\":{\"coefficients\":[-3,1],\"x\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(root_stop, 3, 0, 1, root_found_before_derivative_division, 是)","explanation_zh":"Wave6第十三層核對=(root_stop, 3, 0, 1, root_found_before_derivative_division, 是)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191078,"parameters":{"mode":"root_stop","model":{"coefficients":[-3,1],"x":3},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"root_stop\",\"model\":{\"coefficients\":[[-3,1],[1,1]],\"x\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0079","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":6,\"intercept\":-4,\"slope\":5}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(residual_forward_confusion, 5, 4/5, 26, 26/5, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)","explanation_zh":"Wave6第十三層核對=(residual_forward_confusion, 5, 4/5, 26, 26/5, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191079,"parameters":{"mode":"residual_forward_confusion","model":{"slope":5,"intercept":-4,"approximation":6},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":[6,1],\"intercept\":[-4,1],\"slope\":[5,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0080","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[5,0,1],\"x\":0}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_derivative, 0, 5, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)","explanation_zh":"Wave6第十三層核對=(zero_derivative, 0, 5, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191080,"parameters":{"mode":"zero_derivative","model":{"coefficients":[5,0,1],"x":0},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[[5,1],[0,1],[1,1]],\"x\":[0,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0081","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[-6,0,1],\"steps\":1,\"x0\":2}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191081,"parameters":{"mode":"finite_trace_only","model":{"coefficients":[-6,0,1],"x0":2,"steps":1},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[[-6,1],[0,1],[1,1]],\"steps\":1,\"x0\":[2,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0082","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"root_stop\",\"model\":{\"coefficients\":[-7,1],\"x\":7}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(root_stop, 7, 0, 1, root_found_before_derivative_division, 是)","explanation_zh":"Wave6第十三層核對=(root_stop, 7, 0, 1, root_found_before_derivative_division, 是)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191082,"parameters":{"mode":"root_stop","model":{"coefficients":[-7,1],"x":7},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"root_stop\",\"model\":{\"coefficients\":[[-7,1],[1,1]],\"x\":[7,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0083","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":10,\"intercept\":-8,\"slope\":9}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(residual_forward_confusion, 9, 8/9, 82, 82/9, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)","explanation_zh":"Wave6第十三層核對=(residual_forward_confusion, 9, 8/9, 82, 82/9, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191083,"parameters":{"mode":"residual_forward_confusion","model":{"slope":9,"intercept":-8,"approximation":10},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":[10,1],\"intercept\":[-8,1],\"slope\":[9,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0084","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[9,0,1],\"x\":0}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_derivative, 0, 9, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)","explanation_zh":"Wave6第十三層核對=(zero_derivative, 0, 9, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191084,"parameters":{"mode":"zero_derivative","model":{"coefficients":[9,0,1],"x":0},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[[9,1],[0,1],[1,1]],\"x\":[0,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0085","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[-10,0,1],\"steps\":1,\"x0\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191085,"parameters":{"mode":"finite_trace_only","model":{"coefficients":[-10,0,1],"x0":1,"steps":1},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[[-10,1],[0,1],[1,1]],\"steps\":1,\"x0\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0086","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"root_stop\",\"model\":{\"coefficients\":[-11,1],\"x\":11}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(root_stop, 11, 0, 1, root_found_before_derivative_division, 是)","explanation_zh":"Wave6第十三層核對=(root_stop, 11, 0, 1, root_found_before_derivative_division, 是)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191086,"parameters":{"mode":"root_stop","model":{"coefficients":[-11,1],"x":11},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"root_stop\",\"model\":{\"coefficients\":[[-11,1],[1,1]],\"x\":[11,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0087","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":14,\"intercept\":-12,\"slope\":13}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(residual_forward_confusion, 13, 12/13, 170, 170/13, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)","explanation_zh":"Wave6第十三層核對=(residual_forward_confusion, 13, 12/13, 170, 170/13, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191087,"parameters":{"mode":"residual_forward_confusion","model":{"slope":13,"intercept":-12,"approximation":14},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":[14,1],\"intercept\":[-12,1],\"slope\":[13,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0088","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[13,0,1],\"x\":0}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_derivative, 0, 13, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)","explanation_zh":"Wave6第十三層核對=(zero_derivative, 0, 13, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191088,"parameters":{"mode":"zero_derivative","model":{"coefficients":[13,0,1],"x":0},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[[13,1],[0,1],[1,1]],\"x\":[0,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0089","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[-14,0,1],\"steps\":1,\"x0\":5}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191089,"parameters":{"mode":"finite_trace_only","model":{"coefficients":[-14,0,1],"x0":5,"steps":1},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[[-14,1],[0,1],[1,1]],\"steps\":1,\"x0\":[5,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0090","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"root_stop\",\"model\":{\"coefficients\":[-15,1],\"x\":15}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(root_stop, 15, 0, 1, root_found_before_derivative_division, 是)","explanation_zh":"Wave6第十三層核對=(root_stop, 15, 0, 1, root_found_before_derivative_division, 是)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191090,"parameters":{"mode":"root_stop","model":{"coefficients":[-15,1],"x":15},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"root_stop\",\"model\":{\"coefficients\":[[-15,1],[1,1]],\"x\":[15,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0091","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":18,\"intercept\":-16,\"slope\":17}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(residual_forward_confusion, 17, 16/17, 290, 290/17, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)","explanation_zh":"Wave6第十三層核對=(residual_forward_confusion, 17, 16/17, 290, 290/17, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191091,"parameters":{"mode":"residual_forward_confusion","model":{"slope":17,"intercept":-16,"approximation":18},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":[18,1],\"intercept\":[-16,1],\"slope\":[17,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0092","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[17,0,1],\"x\":0}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_derivative, 0, 17, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)","explanation_zh":"Wave6第十三層核對=(zero_derivative, 0, 17, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191092,"parameters":{"mode":"zero_derivative","model":{"coefficients":[17,0,1],"x":0},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[[17,1],[0,1],[1,1]],\"x\":[0,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0093","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[-18,0,1],\"steps\":1,\"x0\":4}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191093,"parameters":{"mode":"finite_trace_only","model":{"coefficients":[-18,0,1],"x0":4,"steps":1},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[[-18,1],[0,1],[1,1]],\"steps\":1,\"x0\":[4,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0094","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"root_stop\",\"model\":{\"coefficients\":[-19,1],\"x\":19}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(root_stop, 19, 0, 1, root_found_before_derivative_division, 是)","explanation_zh":"Wave6第十三層核對=(root_stop, 19, 0, 1, root_found_before_derivative_division, 是)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191094,"parameters":{"mode":"root_stop","model":{"coefficients":[-19,1],"x":19},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"root_stop\",\"model\":{\"coefficients\":[[-19,1],[1,1]],\"x\":[19,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0095","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":22,\"intercept\":-20,\"slope\":21}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(residual_forward_confusion, 21, 20/21, 442, 442/21, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)","explanation_zh":"Wave6第十三層核對=(residual_forward_confusion, 21, 20/21, 442, 442/21, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191095,"parameters":{"mode":"residual_forward_confusion","model":{"slope":21,"intercept":-20,"approximation":22},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":[22,1],\"intercept\":[-20,1],\"slope\":[21,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0096","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[21,0,1],\"x\":0}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_derivative, 0, 21, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)","explanation_zh":"Wave6第十三層核對=(zero_derivative, 0, 21, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191096,"parameters":{"mode":"zero_derivative","model":{"coefficients":[21,0,1],"x":0},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[[21,1],[0,1],[1,1]],\"x\":[0,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0097","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[-22,0,1],\"steps\":1,\"x0\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191097,"parameters":{"mode":"finite_trace_only","model":{"coefficients":[-22,0,1],"x0":3,"steps":1},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"coefficients\":[[-22,1],[0,1],[1,1]],\"steps\":1,\"x0\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0098","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"root_stop\",\"model\":{\"coefficients\":[-23,1],\"x\":23}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(root_stop, 23, 0, 1, root_found_before_derivative_division, 是)","explanation_zh":"Wave6第十三層核對=(root_stop, 23, 0, 1, root_found_before_derivative_division, 是)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191098,"parameters":{"mode":"root_stop","model":{"coefficients":[-23,1],"x":23},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"root_stop\",\"model\":{\"coefficients\":[[-23,1],[1,1]],\"x\":[23,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0099","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":26,\"intercept\":-24,\"slope\":25}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(residual_forward_confusion, 25, 24/25, 626, 626/25, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)","explanation_zh":"Wave6第十三層核對=(residual_forward_confusion, 25, 24/25, 626, 626/25, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191099,"parameters":{"mode":"residual_forward_confusion","model":{"slope":25,"intercept":-24,"approximation":26},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"residual_forward_confusion\",\"model\":{\"approximation\":[26,1],\"intercept\":[-24,1],\"slope\":[25,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r01-0100","batch_id":"batch-20261015-advance-20261119-r01-newton-method-foundations","language":"zh-Hant","source_mko_ids":["mko-newton-method"],"title_zh":"Newton法","stem_zh":"Newton更新、root-first停止與nonzero derivative責任。操作=newton-boundary；明示資料={\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[25,0,1],\"x\":0}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_derivative, 0, 25, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)","explanation_zh":"Wave6第十三層核對=(zero_derivative, 0, 25, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)；依完整明示資料精確重算；每一步精確重算f與f_prime；root_found先於除法，nonroot zero derivative拒絕；finite orbit或residual下降不證一般收斂、二次rate、global root或forward error。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Newton法"]},"variant":{"family_id":"qf-mod50-newton-boundary","seed":202611191100,"parameters":{"mode":"zero_derivative","model":{"coefficients":[25,0,1],"x":0},"domain":"Exact_Newton_iteration_boundary","operation":"newton-boundary","semantic_template":"newton-boundary|{\"mode\":\"zero_derivative\",\"model\":{\"coefficients\":[[25,1],[0,1],[1,1]],\"x\":[0,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}