第 33909 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":[[9,1],[1,5]],"x":[8,1],"y":[1,-8]}；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":[[9,1],[1,5]],"x":[8,1],"y":[1,-8]}；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, 9, 1]

derived = [[9, 1], 9]

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator := div(9, 1)

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator = 9

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, [9, 1]]

derived = [[9, 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, 5, 1]

derived = [[5, 1], 5]

quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator := div(5, 1)

quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator = 5

source roles = [quadratic Q ORIGINAL row1, [1, 5]]

derived = [[1, 5]]

source roles = [quadratic x ORIGINAL raw coordinate0, 100000000, 8, 1]

derived = [[8, 1], 8]

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(8, 1)

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 8

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, [8, 1]]

derived = [[8, 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, -8, 1]

derived = [[-8, 1], -8]

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(-8, 1)

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator = -8

source roles = [quadratic y ORIGINAL, [1, -8]]

derived = [[1, -8]]

source roles = [Quadratic point difference coordinate0 SOURCE naive sub, 100000000, 7, 1]

derived = [[7, 1], 7]

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator := div(7, 1)

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator = 7

Quadratic point difference coordinate0 ORIGINAL rational sub := sub(8, 1)

Quadratic point difference coordinate0 ORIGINAL rational sub = 7

source roles = [Quadratic point difference coordinate1 SOURCE naive sub, 100000000, 9, 1]

derived = [[9, 1], 9]

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator := div(9, 1)

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator = 9

Quadratic point difference coordinate1 ORIGINAL rational sub := sub(1, -8)

Quadratic point difference coordinate1 ORIGINAL rational sub = 9

source roles = [Quadratic Q times point difference row0 product0 SOURCE naive mul, 100000000, 63, 1]

derived = [[63, 1], 63]

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(63, 1)

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator = 63

Quadratic Q times point difference row0 product0 ORIGINAL rational mul := mul(9, 7)

Quadratic Q times point difference row0 product0 ORIGINAL rational mul = 63

source roles = [Quadratic Q times point difference row0 prefix0 SOURCE naive sum, 100000000, 63, 1]

derived = [[63, 1], 63]

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(63, 1)

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 63

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum := sum(0, 63)

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum = 63

source roles = [Quadratic Q times point difference row0 product1 SOURCE naive mul, 100000000, 9, 1]

derived = [[9, 1], 9]

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(9, 1)

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator = 9

Quadratic Q times point difference row0 product1 ORIGINAL rational mul := mul(1, 9)

Quadratic Q times point difference row0 product1 ORIGINAL rational mul = 9

source roles = [Quadratic Q times point difference row0 prefix1 SOURCE naive sum, 100000000, 72, 1]

derived = [[72, 1], 72]

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(72, 1)

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 72

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum := sum(63, 9)

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum = 72

source roles = [Quadratic Q times point difference row0, [9, 1], [7, 9]]

derived = [72]

source roles = [Quadratic Q times point difference row1 product0 SOURCE naive mul, 100000000, 7, 1]

derived = [[7, 1], 7]

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(7, 1)

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator = 7

Quadratic Q times point difference row1 product0 ORIGINAL rational mul := mul(1, 7)

Quadratic Q times point difference row1 product0 ORIGINAL rational mul = 7

source roles = [Quadratic Q times point difference row1 prefix0 SOURCE naive sum, 100000000, 7, 1]

derived = [[7, 1], 7]

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(7, 1)

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 7

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum := sum(0, 7)

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum = 7

source roles = [Quadratic Q times point difference row1 product1 SOURCE naive mul, 100000000, 45, 1]

derived = [[45, 1], 45]

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(45, 1)

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator = 45

Quadratic Q times point difference row1 product1 ORIGINAL rational mul := mul(5, 9)

Quadratic Q times point difference row1 product1 ORIGINAL rational mul = 45

source roles = [Quadratic Q times point difference row1 prefix1 SOURCE naive sum, 100000000, 52, 1]

derived = [[52, 1], 52]

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(52, 1)

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 52

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum := sum(7, 45)

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum = 52

source roles = [Quadratic Q times point difference row1, [1, 5], [7, 9]]

derived = [52]

source roles = [[[9, 1], [1, 5]], [8, 1], [1, -8]]

derived = [[7, 9], [72, 52]]

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

([[9, 1], [1, 5]], [8, 1], [1, -8], [7, 9], [72, 52], 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。