第 33880 題：梯度向量

原題與已知條件

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[5,-5]}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[5,-5]}；請依 exact_ordered_tuple_declared_by_interface 回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-gradient-vector

逐步推導

source roles = [qf-mod50-graddeep-negative-direction, [domain, gradient, operation]]

derived = [graddeep-negative-direction, Exact_gradient_vector_density]

source roles = [negative direction ORIGINAL gradient raw coordinate0, 100000000, 5, 1]

derived = [[5, 1], 5]

negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator := div(5, 1)

negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator = 5

source roles = [negative direction ORIGINAL gradient raw coordinate1, 100000000, -5, 1]

derived = [[-5, 1], -5]

negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator := div(-5, 1)

negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator = -5

source roles = [negative direction ORIGINAL gradient, [5, -5]]

derived = [[5, -5]]

source roles = [Negative gradient coordinate0 SOURCE naive mul, 100000000, -5, 1]

derived = [[-5, 1], -5]

Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator := div(-5, 1)

Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator = -5

Negative gradient coordinate0 ORIGINAL rational mul := mul(-1, 5)

Negative gradient coordinate0 ORIGINAL rational mul = -5

source roles = [Negative gradient coordinate1 SOURCE naive mul, 100000000, 5, 1]

derived = [[5, 1], 5]

Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator := div(5, 1)

Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator = 5

Negative gradient coordinate1 ORIGINAL rational mul := mul(-1, -5)

Negative gradient coordinate1 ORIGINAL rational mul = 5

source roles = [negative direction inner product product0 SOURCE naive mul, 100000000, -25, 1]

derived = [[-25, 1], -25]

negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator := div(-25, 1)

negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator = -25

negative direction inner product product0 ORIGINAL rational mul := mul(5, -5)

negative direction inner product product0 ORIGINAL rational mul = -25

source roles = [negative direction inner product prefix0 SOURCE naive sum, 100000000, -25, 1]

derived = [[-25, 1], -25]

negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(-25, 1)

negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator = -25

negative direction inner product prefix0 ORIGINAL rational sum := sum(0, -25)

negative direction inner product prefix0 ORIGINAL rational sum = -25

source roles = [negative direction inner product product1 SOURCE naive mul, 100000000, -25, 1]

derived = [[-25, 1], -25]

negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator := div(-25, 1)

negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator = -25

negative direction inner product product1 ORIGINAL rational mul := mul(-5, 5)

negative direction inner product product1 ORIGINAL rational mul = -25

source roles = [negative direction inner product prefix1 SOURCE naive sum, 100000000, -50, 1]

derived = [[-50, 1], -50]

negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(-50, 1)

negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator = -50

negative direction inner product prefix1 ORIGINAL rational sum := sum(-25, -25)

negative direction inner product prefix1 ORIGINAL rational sum = -50

source roles = [negative direction inner product, [5, -5], [-5, 5]]

derived = [-50]

source roles = [negative direction norm squared product0 SOURCE naive mul, 100000000, 25, 1]

derived = [[25, 1], 25]

negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator := div(25, 1)

negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator = 25

negative direction norm squared product0 ORIGINAL rational mul := mul(5, 5)

negative direction norm squared product0 ORIGINAL rational mul = 25

source roles = [negative direction norm squared prefix0 SOURCE naive sum, 100000000, 25, 1]

derived = [[25, 1], 25]

negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(25, 1)

negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 25

negative direction norm squared prefix0 ORIGINAL rational sum := sum(0, 25)

negative direction norm squared prefix0 ORIGINAL rational sum = 25

source roles = [negative direction norm squared product1 SOURCE naive mul, 100000000, 25, 1]

derived = [[25, 1], 25]

negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator := div(25, 1)

negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator = 25

negative direction norm squared product1 ORIGINAL rational mul := mul(-5, -5)

negative direction norm squared product1 ORIGINAL rational mul = 25

source roles = [negative direction norm squared prefix1 SOURCE naive sum, 100000000, 50, 1]

derived = [[50, 1], 50]

negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(50, 1)

negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 50

negative direction norm squared prefix1 ORIGINAL rational sum := sum(25, 25)

negative direction norm squared prefix1 ORIGINAL rational sum = 50

source roles = [negative direction norm squared, [5, -5], [5, -5]]

derived = [50]

source roles = [[5, -5]]

derived = [[-5, 5], -50, 50, 是]

逐步反向核對

• S039 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S038 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S037 · premise_and_result_audit — 以精確有理數重算 negative direction norm squared prefix1 ORIGINAL rational sum，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S035 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S034 · premise_and_result_audit — 以精確有理數重算 negative direction norm squared product1 ORIGINAL rational mul，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S032 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S031 · premise_and_result_audit — 以精確有理數重算 negative direction norm squared prefix0 ORIGINAL rational sum，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S029 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S028 · premise_and_result_audit — 以精確有理數重算 negative direction norm squared product0 ORIGINAL rational mul，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S026 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S025 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S024 · premise_and_result_audit — 以精確有理數重算 negative direction inner product prefix1 ORIGINAL rational sum，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S022 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S021 · premise_and_result_audit — 以精確有理數重算 negative direction inner product product1 ORIGINAL rational mul，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S019 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S018 · premise_and_result_audit — 以精確有理數重算 negative direction inner product prefix0 ORIGINAL rational sum，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S016 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S015 · premise_and_result_audit — 以精確有理數重算 negative direction inner product product0 ORIGINAL rational mul，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S013 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S012 · premise_and_result_audit — 以精確有理數重算 Negative gradient coordinate1 ORIGINAL rational mul，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S010 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S009 · premise_and_result_audit — 以精確有理數重算 Negative gradient coordinate0 ORIGINAL rational mul，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S007 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S006 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S005 · premise_and_result_audit — 以精確有理數重算 negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator，不以小數近似替代。
• S004 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S003 · premise_and_result_audit — 以精確有理數重算 negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator，不以小數近似替代。
• S002 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S001 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。

結論

([5, -5], [-5, 5], -50, 50, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。