# Newton根求解方法基礎 · 脈衝 R01

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261015-advance-20261119-r01-newton-method-foundations · 2026-10-15 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

[欄序與來源綁定資料](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-1, 0, 1], 2, 3, 4, updated, 5/4, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0001/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 2. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-2, 0, 1], 3, 7, 6, updated, 11/6, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0002/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 3. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-3, 0, 1], 4, 13, 8, updated, 19/8, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0003/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 4. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-4, 0, 1], 5, 21, 10, updated, 29/10, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0004/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 5. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-5, 0, 1], 1, -4, 2, updated, 3, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0005/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 6. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-6, 0, 1], 2, -2, 4, updated, 5/2, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0006/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 7. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-7, 0, 1], 3, 2, 6, updated, 8/3, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0007/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 8. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-8, 0, 1], 4, 8, 8, updated, 3, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0008/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 9. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-9, 0, 1], 5, 16, 10, updated, 17/5, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0009/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 10. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-10, 0, 1], 1, -9, 2, updated, 11/2, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0010/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 11. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-11, 0, 1], 2, -7, 4, updated, 15/4, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0011/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 12. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-12, 0, 1], 3, -3, 6, updated, 7/2, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0012/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 13. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-13, 0, 1], 4, 3, 8, updated, 29/8, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0013/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 14. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-14, 0, 1], 5, 11, 10, updated, 39/10, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0014/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 15. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-15, 0, 1], 1, -14, 2, updated, 8, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0015/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 16. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-16, 0, 1], 2, -12, 4, updated, 5, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0016/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 17. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-17, 0, 1], 3, -8, 6, updated, 13/3, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0017/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 18. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-18, 0, 1], 4, -2, 8, updated, 17/4, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0018/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 19. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-19, 0, 1], 5, 6, 10, updated, 22/5, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0019/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 20. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-20, 0, 1], 1, -19, 2, updated, 21/2, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0020/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 21. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-21, 0, 1], 2, -17, 4, updated, 25/4, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0021/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 22. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-22, 0, 1], 3, -13, 6, updated, 31/6, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0022/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 23. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-23, 0, 1], 4, -7, 8, updated, 39/8, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0023/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 24. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-24, 0, 1], 5, 1, 10, updated, 49/10, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0024/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 25. Newton法

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 [coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention] 回答。

先驗f(x)=0而停止；即使f_prime同時為0，也不做0/0。非根且導數為0才拒絕更新。coefficients按低次到高次排列。

查看答案與解釋([-25, 0, 1], 1, -24, 2, updated, 13, root_test_precedes_division)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0025/)

Newton根求解方法 · qf-mod50-newton-step · candidate





## 26. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0026/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 27. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0027/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 28. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0028/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 29. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0029/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 30. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0030/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 31. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0031/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 32. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0032/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 33. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0033/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 34. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0034/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 35. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0035/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 36. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0036/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 37. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0037/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 38. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0038/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 39. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0039/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 40. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0040/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 41. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0041/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 42. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0042/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 43. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0043/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 44. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0044/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 45. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0045/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 46. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0046/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 47. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0047/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 48. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0048/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 49. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0049/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 50. Newton法

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 [coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope] 回答。

step_budget是允許嘗試的步數，不保證records等長；遇exact root提前停止。有限trace與residual不能代替一般收斂或rate證明。

查看答案與解釋([-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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0050/)

Newton根求解方法 · qf-mod50-newton-finite-orbit · candidate





## 51. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0051/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 52. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0052/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 53. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0053/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 54. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0054/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 55. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0055/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 56. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0056/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 57. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0057/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 58. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0058/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 59. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0059/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 60. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0060/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 61. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0061/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 62. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0062/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 63. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0063/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 64. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0064/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 65. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0065/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 66. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0066/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 67. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0067/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 68. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0068/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 69. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0069/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 70. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0070/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 71. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0071/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 72. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0072/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 73. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0073/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 74. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0074/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 75. Newton法

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 [radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope] 回答。

查看答案與解釋(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)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0075/)

Newton根求解方法 · qf-mod50-newton-square-root-orbit · candidate





## 76. Newton法

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 [mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(zero_derivative, 0, 1, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0076/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 77. Newton法

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 [mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(finite_trace_only, 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0077/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 78. Newton法

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 [mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(root_stop, 3, 0, 1, root_found_before_derivative_division, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0078/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 79. Newton法

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 [mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(residual_forward_confusion, 5, 4/5, 26, 26/5, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0079/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 80. Newton法

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 [mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(zero_derivative, 0, 5, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0080/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 81. Newton法

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 [mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(finite_trace_only, 1, [[0, 2, -2, 4, updated, 5/2]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0081/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 82. Newton法

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 [mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(root_stop, 7, 0, 1, root_found_before_derivative_division, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0082/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 83. Newton法

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 [mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(residual_forward_confusion, 9, 8/9, 82, 82/9, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0083/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 84. Newton法

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 [mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(zero_derivative, 0, 9, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0084/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 85. Newton法

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 [mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(finite_trace_only, 1, [[0, 1, -9, 2, updated, 11/2]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0085/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 86. Newton法

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 [mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(root_stop, 11, 0, 1, root_found_before_derivative_division, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0086/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 87. Newton法

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 [mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(residual_forward_confusion, 13, 12/13, 170, 170/13, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0087/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 88. Newton法

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 [mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(zero_derivative, 0, 13, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0088/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 89. Newton法

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 [mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(finite_trace_only, 1, [[0, 5, 11, 10, updated, 39/10]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0089/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 90. Newton法

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 [mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(root_stop, 15, 0, 1, root_found_before_derivative_division, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0090/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 91. Newton法

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 [mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(residual_forward_confusion, 17, 16/17, 290, 290/17, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0091/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 92. Newton法

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 [mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(zero_derivative, 0, 17, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0092/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 93. Newton法

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 [mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(finite_trace_only, 1, [[0, 4, -2, 8, updated, 17/4]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0093/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 94. Newton法

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 [mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(root_stop, 19, 0, 1, root_found_before_derivative_division, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0094/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 95. Newton法

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 [mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(residual_forward_confusion, 21, 20/21, 442, 442/21, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0095/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 96. Newton法

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 [mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(zero_derivative, 0, 21, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0096/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 97. Newton法

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 [mode, step_budget, step_records, status, convergence_or_rate_proof_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(finite_trace_only, 1, [[0, 3, -13, 6, updated, 31/6]], step_budget_exhausted, 否, finite_residual_behavior_does_not_prove_convergence_or_rate)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0097/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 98. Newton法

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 [mode, current_iterate, function_value, derivative_value, root_first_status, root_stop_qualified] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(root_stop, 23, 0, 1, root_found_before_derivative_division, 是)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0098/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 99. Newton法

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 [mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(residual_forward_confusion, 25, 24/25, 626, 626/25, 否, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0099/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate





## 100. Newton法

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 [mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary] 回答。

四種mode為7/6/6/7欄。residual_equals_forward_error是本次數值相等旗標，不保證所有affine slopes都相等；正確關係含abs(slope)。

查看答案與解釋(zero_derivative, 0, 25, 0, undefined_zero_derivative, 否, division_by_zero_is_rejected_not_repaired)

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。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r01-0100/)

Newton根求解方法 · qf-mod50-newton-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r01-newton-method-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r01-newton-method-foundations/validation-report.json)