第 33925 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-quadratic-difference；明示資料={"symmetric_matrix":[[25,1],[1,6]],"x":[24,1],"y":[1,-24]}；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-quadratic-difference；明示資料={"symmetric_matrix":[[25,1],[1,6]],"x":[24,1],"y":[1,-24]}；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-quadratic-difference, [domain, operation, symmetric_matrix, x, y]]

derived = [graddeep-quadratic-difference, Exact_gradient_vector_density]

source roles = [quadratic Q ORIGINAL row0 raw coordinate0, 100000000, 25, 1]

derived = [[25, 1], 25]

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator := div(25, 1)

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator = 25

source roles = [quadratic Q ORIGINAL row0 raw coordinate1, 100000000, 1, 1]

derived = [[1, 1], 1]

quadratic Q ORIGINAL row0 raw coordinate1 ORIGINAL numerator over denominator := div(1, 1)

quadratic Q ORIGINAL row0 raw coordinate1 ORIGINAL numerator over denominator = 1

source roles = [quadratic Q ORIGINAL row0, [25, 1]]

derived = [[25, 1]]

source roles = [quadratic Q ORIGINAL row1 raw coordinate0, 100000000, 1, 1]

derived = [[1, 1], 1]

quadratic Q ORIGINAL row1 raw coordinate0 ORIGINAL numerator over denominator := div(1, 1)

quadratic Q ORIGINAL row1 raw coordinate0 ORIGINAL numerator over denominator = 1

source roles = [quadratic Q ORIGINAL row1 raw coordinate1, 100000000, 6, 1]

derived = [[6, 1], 6]

quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator := div(6, 1)

quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator = 6

source roles = [quadratic Q ORIGINAL row1, [1, 6]]

derived = [[1, 6]]

source roles = [quadratic x ORIGINAL raw coordinate0, 100000000, 24, 1]

derived = [[24, 1], 24]

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(24, 1)

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 24

source roles = [quadratic x ORIGINAL raw coordinate1, 100000000, 1, 1]

derived = [[1, 1], 1]

quadratic x ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(1, 1)

quadratic x ORIGINAL raw coordinate1 ORIGINAL numerator over denominator = 1

source roles = [quadratic x ORIGINAL, [24, 1]]

derived = [[24, 1]]

source roles = [quadratic y ORIGINAL raw coordinate0, 100000000, 1, 1]

derived = [[1, 1], 1]

quadratic y ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(1, 1)

quadratic y ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 1

source roles = [quadratic y ORIGINAL raw coordinate1, 100000000, -24, 1]

derived = [[-24, 1], -24]

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(-24, 1)

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator = -24

source roles = [quadratic y ORIGINAL, [1, -24]]

derived = [[1, -24]]

source roles = [Quadratic point difference coordinate0 SOURCE naive sub, 100000000, 23, 1]

derived = [[23, 1], 23]

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator := div(23, 1)

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator = 23

Quadratic point difference coordinate0 ORIGINAL rational sub := sub(24, 1)

Quadratic point difference coordinate0 ORIGINAL rational sub = 23

source roles = [Quadratic point difference coordinate1 SOURCE naive sub, 100000000, 25, 1]

derived = [[25, 1], 25]

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator := div(25, 1)

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator = 25

Quadratic point difference coordinate1 ORIGINAL rational sub := sub(1, -24)

Quadratic point difference coordinate1 ORIGINAL rational sub = 25

source roles = [Quadratic Q times point difference row0 product0 SOURCE naive mul, 100000000, 575, 1]

derived = [[575, 1], 575]

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(575, 1)

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator = 575

Quadratic Q times point difference row0 product0 ORIGINAL rational mul := mul(25, 23)

Quadratic Q times point difference row0 product0 ORIGINAL rational mul = 575

source roles = [Quadratic Q times point difference row0 prefix0 SOURCE naive sum, 100000000, 575, 1]

derived = [[575, 1], 575]

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(575, 1)

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 575

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum := sum(0, 575)

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum = 575

source roles = [Quadratic Q times point difference row0 product1 SOURCE naive mul, 100000000, 25, 1]

derived = [[25, 1], 25]

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(25, 1)

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator = 25

Quadratic Q times point difference row0 product1 ORIGINAL rational mul := mul(1, 25)

Quadratic Q times point difference row0 product1 ORIGINAL rational mul = 25

source roles = [Quadratic Q times point difference row0 prefix1 SOURCE naive sum, 100000000, 600, 1]

derived = [[600, 1], 600]

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(600, 1)

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 600

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum := sum(575, 25)

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum = 600

source roles = [Quadratic Q times point difference row0, [25, 1], [23, 25]]

derived = [600]

source roles = [Quadratic Q times point difference row1 product0 SOURCE naive mul, 100000000, 23, 1]

derived = [[23, 1], 23]

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(23, 1)

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator = 23

Quadratic Q times point difference row1 product0 ORIGINAL rational mul := mul(1, 23)

Quadratic Q times point difference row1 product0 ORIGINAL rational mul = 23

source roles = [Quadratic Q times point difference row1 prefix0 SOURCE naive sum, 100000000, 23, 1]

derived = [[23, 1], 23]

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(23, 1)

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 23

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum := sum(0, 23)

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum = 23

source roles = [Quadratic Q times point difference row1 product1 SOURCE naive mul, 100000000, 150, 1]

derived = [[150, 1], 150]

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(150, 1)

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator = 150

Quadratic Q times point difference row1 product1 ORIGINAL rational mul := mul(6, 25)

Quadratic Q times point difference row1 product1 ORIGINAL rational mul = 150

source roles = [Quadratic Q times point difference row1 prefix1 SOURCE naive sum, 100000000, 173, 1]

derived = [[173, 1], 173]

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(173, 1)

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 173

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum := sum(23, 150)

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum = 173

source roles = [Quadratic Q times point difference row1, [1, 6], [23, 25]]

derived = [173]

source roles = [[[25, 1], [1, 6]], [24, 1], [1, -24]]

derived = [[23, 25], [600, 173]]

逐步反向核對

• S054 · 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。
• S053 · 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。
• S052 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S050 · 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。
• S049 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row1 product1 ORIGINAL rational mul，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S044 · 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。
• S043 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row1 product0 ORIGINAL rational mul，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S041 · 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。
• S040 · 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。
• S039 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S037 · 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。
• S036 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row0 product1 ORIGINAL rational mul，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S034 · 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。
• S033 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S031 · 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。
• S030 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row0 product0 ORIGINAL rational mul，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S028 · 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。
• S027 · premise_and_result_audit — 以精確有理數重算 Quadratic point difference coordinate1 ORIGINAL rational sub，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator，不以小數近似替代。
• 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 Quadratic point difference coordinate0 ORIGINAL rational sub，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Quadratic point difference coordinate0 SOURCE naive sub 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 — 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。
• S020 · premise_and_result_audit — 以精確有理數重算 quadratic y ORIGINAL raw coordinate1 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 quadratic y ORIGINAL raw coordinate0 ORIGINAL numerator over denominator，不以小數近似替代。
• S017 · 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。
• 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 quadratic x ORIGINAL raw coordinate1 ORIGINAL numerator over denominator，不以小數近似替代。
• S014 · 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。
• S013 · premise_and_result_audit — 以精確有理數重算 quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator，不以小數近似替代。
• S012 · 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。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator，不以小數近似替代。
• S009 · 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。
• S008 · premise_and_result_audit — 以精確有理數重算 quadratic Q ORIGINAL row1 raw coordinate0 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 quadratic Q ORIGINAL row0 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 quadratic Q ORIGINAL row0 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

([[25, 1], [1, 6]], [24, 1], [1, -24], [23, 25], [600, 173], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

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。