第 33915 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":[[15,1],[1,6]],"x":[14,1],"y":[1,-14]}；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":[[15,1],[1,6]],"x":[14,1],"y":[1,-14]}；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, 15, 1]

derived = [[15, 1], 15]

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator := div(15, 1)

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator = 15

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, [15, 1]]

derived = [[15, 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, 14, 1]

derived = [[14, 1], 14]

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(14, 1)

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 14

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, [14, 1]]

derived = [[14, 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, -14, 1]

derived = [[-14, 1], -14]

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(-14, 1)

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator = -14

source roles = [quadratic y ORIGINAL, [1, -14]]

derived = [[1, -14]]

source roles = [Quadratic point difference coordinate0 SOURCE naive sub, 100000000, 13, 1]

derived = [[13, 1], 13]

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator := div(13, 1)

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator = 13

Quadratic point difference coordinate0 ORIGINAL rational sub := sub(14, 1)

Quadratic point difference coordinate0 ORIGINAL rational sub = 13

source roles = [Quadratic point difference coordinate1 SOURCE naive sub, 100000000, 15, 1]

derived = [[15, 1], 15]

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator := div(15, 1)

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator = 15

Quadratic point difference coordinate1 ORIGINAL rational sub := sub(1, -14)

Quadratic point difference coordinate1 ORIGINAL rational sub = 15

source roles = [Quadratic Q times point difference row0 product0 SOURCE naive mul, 100000000, 195, 1]

derived = [[195, 1], 195]

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(195, 1)

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator = 195

Quadratic Q times point difference row0 product0 ORIGINAL rational mul := mul(15, 13)

Quadratic Q times point difference row0 product0 ORIGINAL rational mul = 195

source roles = [Quadratic Q times point difference row0 prefix0 SOURCE naive sum, 100000000, 195, 1]

derived = [[195, 1], 195]

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(195, 1)

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 195

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum := sum(0, 195)

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum = 195

source roles = [Quadratic Q times point difference row0 product1 SOURCE naive mul, 100000000, 15, 1]

derived = [[15, 1], 15]

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(15, 1)

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator = 15

Quadratic Q times point difference row0 product1 ORIGINAL rational mul := mul(1, 15)

Quadratic Q times point difference row0 product1 ORIGINAL rational mul = 15

source roles = [Quadratic Q times point difference row0 prefix1 SOURCE naive sum, 100000000, 210, 1]

derived = [[210, 1], 210]

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(210, 1)

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 210

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum := sum(195, 15)

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum = 210

source roles = [Quadratic Q times point difference row0, [15, 1], [13, 15]]

derived = [210]

source roles = [Quadratic Q times point difference row1 product0 SOURCE naive mul, 100000000, 13, 1]

derived = [[13, 1], 13]

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(13, 1)

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator = 13

Quadratic Q times point difference row1 product0 ORIGINAL rational mul := mul(1, 13)

Quadratic Q times point difference row1 product0 ORIGINAL rational mul = 13

source roles = [Quadratic Q times point difference row1 prefix0 SOURCE naive sum, 100000000, 13, 1]

derived = [[13, 1], 13]

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(13, 1)

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 13

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum := sum(0, 13)

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum = 13

source roles = [Quadratic Q times point difference row1 product1 SOURCE naive mul, 100000000, 90, 1]

derived = [[90, 1], 90]

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(90, 1)

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator = 90

Quadratic Q times point difference row1 product1 ORIGINAL rational mul := mul(6, 15)

Quadratic Q times point difference row1 product1 ORIGINAL rational mul = 90

source roles = [Quadratic Q times point difference row1 prefix1 SOURCE naive sum, 100000000, 103, 1]

derived = [[103, 1], 103]

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(103, 1)

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 103

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum := sum(13, 90)

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum = 103

source roles = [Quadratic Q times point difference row1, [1, 6], [13, 15]]

derived = [103]

source roles = [[[15, 1], [1, 6]], [14, 1], [1, -14]]

derived = [[13, 15], [210, 103]]

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

([[15, 1], [1, 6]], [14, 1], [1, -14], [13, 15], [210, 103], 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。