第 33921 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":[[21,1],[1,2]],"x":[20,1],"y":[1,-20]}；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":[[21,1],[1,2]],"x":[20,1],"y":[1,-20]}；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, 21, 1]

derived = [[21, 1], 21]

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator := div(21, 1)

quadratic Q ORIGINAL row0 raw coordinate0 ORIGINAL numerator over denominator = 21

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, [21, 1]]

derived = [[21, 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, 2, 1]

derived = [[2, 1], 2]

quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator := div(2, 1)

quadratic Q ORIGINAL row1 raw coordinate1 ORIGINAL numerator over denominator = 2

source roles = [quadratic Q ORIGINAL row1, [1, 2]]

derived = [[1, 2]]

source roles = [quadratic x ORIGINAL raw coordinate0, 100000000, 20, 1]

derived = [[20, 1], 20]

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator := div(20, 1)

quadratic x ORIGINAL raw coordinate0 ORIGINAL numerator over denominator = 20

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, [20, 1]]

derived = [[20, 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, -20, 1]

derived = [[-20, 1], -20]

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator := div(-20, 1)

quadratic y ORIGINAL raw coordinate1 ORIGINAL numerator over denominator = -20

source roles = [quadratic y ORIGINAL, [1, -20]]

derived = [[1, -20]]

source roles = [Quadratic point difference coordinate0 SOURCE naive sub, 100000000, 19, 1]

derived = [[19, 1], 19]

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator := div(19, 1)

Quadratic point difference coordinate0 SOURCE naive sub ORIGINAL numerator over denominator = 19

Quadratic point difference coordinate0 ORIGINAL rational sub := sub(20, 1)

Quadratic point difference coordinate0 ORIGINAL rational sub = 19

source roles = [Quadratic point difference coordinate1 SOURCE naive sub, 100000000, 21, 1]

derived = [[21, 1], 21]

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator := div(21, 1)

Quadratic point difference coordinate1 SOURCE naive sub ORIGINAL numerator over denominator = 21

Quadratic point difference coordinate1 ORIGINAL rational sub := sub(1, -20)

Quadratic point difference coordinate1 ORIGINAL rational sub = 21

source roles = [Quadratic Q times point difference row0 product0 SOURCE naive mul, 100000000, 399, 1]

derived = [[399, 1], 399]

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(399, 1)

Quadratic Q times point difference row0 product0 SOURCE naive mul ORIGINAL numerator over denominator = 399

Quadratic Q times point difference row0 product0 ORIGINAL rational mul := mul(21, 19)

Quadratic Q times point difference row0 product0 ORIGINAL rational mul = 399

source roles = [Quadratic Q times point difference row0 prefix0 SOURCE naive sum, 100000000, 399, 1]

derived = [[399, 1], 399]

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(399, 1)

Quadratic Q times point difference row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 399

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum := sum(0, 399)

Quadratic Q times point difference row0 prefix0 ORIGINAL rational sum = 399

source roles = [Quadratic Q times point difference row0 product1 SOURCE naive mul, 100000000, 21, 1]

derived = [[21, 1], 21]

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(21, 1)

Quadratic Q times point difference row0 product1 SOURCE naive mul ORIGINAL numerator over denominator = 21

Quadratic Q times point difference row0 product1 ORIGINAL rational mul := mul(1, 21)

Quadratic Q times point difference row0 product1 ORIGINAL rational mul = 21

source roles = [Quadratic Q times point difference row0 prefix1 SOURCE naive sum, 100000000, 420, 1]

derived = [[420, 1], 420]

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(420, 1)

Quadratic Q times point difference row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 420

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum := sum(399, 21)

Quadratic Q times point difference row0 prefix1 ORIGINAL rational sum = 420

source roles = [Quadratic Q times point difference row0, [21, 1], [19, 21]]

derived = [420]

source roles = [Quadratic Q times point difference row1 product0 SOURCE naive mul, 100000000, 19, 1]

derived = [[19, 1], 19]

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(19, 1)

Quadratic Q times point difference row1 product0 SOURCE naive mul ORIGINAL numerator over denominator = 19

Quadratic Q times point difference row1 product0 ORIGINAL rational mul := mul(1, 19)

Quadratic Q times point difference row1 product0 ORIGINAL rational mul = 19

source roles = [Quadratic Q times point difference row1 prefix0 SOURCE naive sum, 100000000, 19, 1]

derived = [[19, 1], 19]

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(19, 1)

Quadratic Q times point difference row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 19

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum := sum(0, 19)

Quadratic Q times point difference row1 prefix0 ORIGINAL rational sum = 19

source roles = [Quadratic Q times point difference row1 product1 SOURCE naive mul, 100000000, 42, 1]

derived = [[42, 1], 42]

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(42, 1)

Quadratic Q times point difference row1 product1 SOURCE naive mul ORIGINAL numerator over denominator = 42

Quadratic Q times point difference row1 product1 ORIGINAL rational mul := mul(2, 21)

Quadratic Q times point difference row1 product1 ORIGINAL rational mul = 42

source roles = [Quadratic Q times point difference row1 prefix1 SOURCE naive sum, 100000000, 61, 1]

derived = [[61, 1], 61]

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(61, 1)

Quadratic Q times point difference row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 61

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum := sum(19, 42)

Quadratic Q times point difference row1 prefix1 ORIGINAL rational sum = 61

source roles = [Quadratic Q times point difference row1, [1, 2], [19, 21]]

derived = [61]

source roles = [[[21, 1], [1, 2]], [20, 1], [1, -20]]

derived = [[19, 21], [420, 61]]

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

([[21, 1], [1, 2]], [20, 1], [1, -20], [19, 21], [420, 61], 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。