第 33832 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-boundary；明示資料={"mode":"non_euclidean_metric","model":{"coordinate_gradient":[7,1],"metric":[[2,0],[0,3]]}}；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-boundary；明示資料={"mode":"non_euclidean_metric","model":{"coordinate_gradient":[7,1],"metric":[[2,0],[0,3]]}}；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-boundary, [domain, mode, model, operation]]

derived = [graddeep-boundary, Exact_gradient_vector_density]

source roles = [metric ORIGINAL covector raw coordinate0, 100000000, 7, 1]

derived = [[7, 1], 7]

metric ORIGINAL covector raw coordinate0 ORIGINAL numerator over denominator := div(7, 1)

metric ORIGINAL covector raw coordinate0 ORIGINAL numerator over denominator = 7

source roles = [metric ORIGINAL covector raw coordinate1, 100000000, 1, 1]

derived = [[1, 1], 1]

metric ORIGINAL covector raw coordinate1 ORIGINAL numerator over denominator := div(1, 1)

metric ORIGINAL covector raw coordinate1 ORIGINAL numerator over denominator = 1

source roles = [metric ORIGINAL covector, [7, 1]]

derived = [[7, 1]]

source roles = [metric ORIGINAL matrix row0 raw coordinate0, 100000000, 2, 1]

derived = [[2, 1], 2]

metric ORIGINAL matrix row0 raw coordinate0 ORIGINAL numerator over denominator := div(2, 1)

metric ORIGINAL matrix row0 raw coordinate0 ORIGINAL numerator over denominator = 2

source roles = [metric ORIGINAL matrix row0 raw coordinate1, 100000000, 0, 1]

derived = [[0, 1], 0]

metric ORIGINAL matrix row0 raw coordinate1 ORIGINAL numerator over denominator := div(0, 1)

metric ORIGINAL matrix row0 raw coordinate1 ORIGINAL numerator over denominator = 0

source roles = [metric ORIGINAL matrix row0, [2, 0]]

derived = [[2, 0]]

source roles = [metric ORIGINAL matrix row1 raw coordinate0, 100000000, 0, 1]

derived = [[0, 1], 0]

metric ORIGINAL matrix row1 raw coordinate0 ORIGINAL numerator over denominator := div(0, 1)

metric ORIGINAL matrix row1 raw coordinate0 ORIGINAL numerator over denominator = 0

source roles = [metric ORIGINAL matrix row1 raw coordinate1, 100000000, 3, 1]

derived = [[3, 1], 3]

metric ORIGINAL matrix row1 raw coordinate1 ORIGINAL numerator over denominator := div(3, 1)

metric ORIGINAL matrix row1 raw coordinate1 ORIGINAL numerator over denominator = 3

source roles = [metric ORIGINAL matrix row1, [0, 3]]

derived = [[0, 3]]

source roles = [Metric determinant main SOURCE naive mul, 100000000, 6, 1]

derived = [[6, 1], 6]

Metric determinant main SOURCE naive mul ORIGINAL numerator over denominator := div(6, 1)

Metric determinant main SOURCE naive mul ORIGINAL numerator over denominator = 6

Metric determinant main ORIGINAL rational mul := mul(2, 3)

Metric determinant main ORIGINAL rational mul = 6

source roles = [Metric determinant cross SOURCE naive mul, 100000000, 0, 1]

derived = [[0, 1], 0]

Metric determinant cross SOURCE naive mul ORIGINAL numerator over denominator := div(0, 1)

Metric determinant cross SOURCE naive mul ORIGINAL numerator over denominator = 0

Metric determinant cross ORIGINAL rational mul := mul(0, 0)

Metric determinant cross ORIGINAL rational mul = 0

source roles = [Metric determinant signed SOURCE naive sub, 100000000, 6, 1]

derived = [[6, 1], 6]

Metric determinant signed SOURCE naive sub ORIGINAL numerator over denominator := div(6, 1)

Metric determinant signed SOURCE naive sub ORIGINAL numerator over denominator = 6

Metric determinant signed ORIGINAL rational sub := sub(6, 0)

Metric determinant signed ORIGINAL rational sub = 6

source roles = [Metric inverse00 SOURCE naive div, 100000000, 3, 6]

derived = [[1, 2], 1/2]

Metric inverse00 SOURCE naive div ORIGINAL numerator over denominator := div(3, 6)

Metric inverse00 SOURCE naive div ORIGINAL numerator over denominator = 1/2

Metric inverse00 ORIGINAL rational div := div(3, 6)

Metric inverse00 ORIGINAL rational div = 1/2

source roles = [Metric inverse01 SOURCE naive div, 100000000, 0, 6]

derived = [[0, 1], 0]

Metric inverse01 SOURCE naive div ORIGINAL numerator over denominator := div(0, 6)

Metric inverse01 SOURCE naive div ORIGINAL numerator over denominator = 0

Metric inverse01 ORIGINAL rational div := div(0, 6)

Metric inverse01 ORIGINAL rational div = 0

source roles = [Metric inverse10 SOURCE naive div, 100000000, 0, 6]

derived = [[0, 1], 0]

Metric inverse10 SOURCE naive div ORIGINAL numerator over denominator := div(0, 6)

Metric inverse10 SOURCE naive div ORIGINAL numerator over denominator = 0

Metric inverse10 ORIGINAL rational div := div(0, 6)

Metric inverse10 ORIGINAL rational div = 0

source roles = [Metric inverse11 SOURCE naive div, 100000000, 2, 6]

derived = [[1, 3], 1/3]

Metric inverse11 SOURCE naive div ORIGINAL numerator over denominator := div(2, 6)

Metric inverse11 SOURCE naive div ORIGINAL numerator over denominator = 1/3

Metric inverse11 ORIGINAL rational div := div(2, 6)

Metric inverse11 ORIGINAL rational div = 1/3

source roles = [Metric inverse times covector row0 product0 SOURCE naive mul, 100000000, 7, 2]

derived = [[7, 2], 7/2]

Metric inverse times covector row0 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(7, 2)

Metric inverse times covector row0 product0 SOURCE naive mul ORIGINAL numerator over denominator = 7/2

Metric inverse times covector row0 product0 ORIGINAL rational mul := mul(1/2, 7)

Metric inverse times covector row0 product0 ORIGINAL rational mul = 7/2

source roles = [Metric inverse times covector row0 prefix0 SOURCE naive sum, 100000000, 7, 2]

derived = [[7, 2], 7/2]

Metric inverse times covector row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(7, 2)

Metric inverse times covector row0 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 7/2

Metric inverse times covector row0 prefix0 ORIGINAL rational sum := sum(0, 7/2)

Metric inverse times covector row0 prefix0 ORIGINAL rational sum = 7/2

source roles = [Metric inverse times covector row0 product1 SOURCE naive mul, 100000000, 0, 1]

derived = [[0, 1], 0]

Metric inverse times covector row0 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(0, 1)

Metric inverse times covector row0 product1 SOURCE naive mul ORIGINAL numerator over denominator = 0

Metric inverse times covector row0 product1 ORIGINAL rational mul := mul(0, 1)

Metric inverse times covector row0 product1 ORIGINAL rational mul = 0

source roles = [Metric inverse times covector row0 prefix1 SOURCE naive sum, 100000000, 7, 2]

derived = [[7, 2], 7/2]

Metric inverse times covector row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(7, 2)

Metric inverse times covector row0 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 7/2

Metric inverse times covector row0 prefix1 ORIGINAL rational sum := sum(7/2, 0)

Metric inverse times covector row0 prefix1 ORIGINAL rational sum = 7/2

source roles = [Metric inverse times covector row0, [1/2, 0], [7, 1]]

derived = [7/2]

source roles = [Metric inverse times covector row1 product0 SOURCE naive mul, 100000000, 0, 1]

derived = [[0, 1], 0]

Metric inverse times covector row1 product0 SOURCE naive mul ORIGINAL numerator over denominator := div(0, 1)

Metric inverse times covector row1 product0 SOURCE naive mul ORIGINAL numerator over denominator = 0

Metric inverse times covector row1 product0 ORIGINAL rational mul := mul(0, 7)

Metric inverse times covector row1 product0 ORIGINAL rational mul = 0

source roles = [Metric inverse times covector row1 prefix0 SOURCE naive sum, 100000000, 0, 1]

derived = [[0, 1], 0]

Metric inverse times covector row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(0, 1)

Metric inverse times covector row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 0

Metric inverse times covector row1 prefix0 ORIGINAL rational sum := sum(0, 0)

Metric inverse times covector row1 prefix0 ORIGINAL rational sum = 0

source roles = [Metric inverse times covector row1 product1 SOURCE naive mul, 100000000, 1, 3]

derived = [[1, 3], 1/3]

Metric inverse times covector row1 product1 SOURCE naive mul ORIGINAL numerator over denominator := div(1, 3)

Metric inverse times covector row1 product1 SOURCE naive mul ORIGINAL numerator over denominator = 1/3

Metric inverse times covector row1 product1 ORIGINAL rational mul := mul(1/3, 1)

Metric inverse times covector row1 product1 ORIGINAL rational mul = 1/3

source roles = [Metric inverse times covector row1 prefix1 SOURCE naive sum, 100000000, 1, 3]

derived = [[1, 3], 1/3]

Metric inverse times covector row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(1, 3)

Metric inverse times covector row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 1/3

Metric inverse times covector row1 prefix1 ORIGINAL rational sum := sum(0, 1/3)

Metric inverse times covector row1 prefix1 ORIGINAL rational sum = 1/3

source roles = [Metric inverse times covector row1, [0, 1/3], [7, 1]]

derived = [1/3]

source roles = [[7, 1], [[2, 0], [0, 3]], 6]

derived = [[[1/2, 0], [0, 1/3]], [7/2, 1/3], 否]

逐步反向核對

• S064 · 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。
• S063 · 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。
• S062 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row1 prefix1 ORIGINAL rational sum，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row1 prefix1 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S060 · 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。
• S059 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row1 product1 ORIGINAL rational mul，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row1 product1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S057 · 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。
• S056 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row1 prefix0 ORIGINAL rational sum，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row1 prefix0 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• 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 — 以精確有理數重算 Metric inverse times covector row1 product0 ORIGINAL rational mul，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row1 product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S051 · 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。
• 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 Metric inverse times covector row0 prefix1 ORIGINAL rational sum，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row0 prefix1 SOURCE naive sum 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 Metric inverse times covector row0 product1 ORIGINAL rational mul，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row0 product1 SOURCE naive mul 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 Metric inverse times covector row0 prefix0 ORIGINAL rational sum，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row0 prefix0 SOURCE naive sum 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 — 以精確有理數重算 Metric inverse times covector row0 product0 ORIGINAL rational mul，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Metric inverse times covector row0 product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• 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 Metric inverse11 ORIGINAL rational div，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Metric inverse11 SOURCE naive div 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 Metric inverse10 ORIGINAL rational div，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Metric inverse10 SOURCE naive div 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 Metric inverse01 ORIGINAL rational div，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Metric inverse01 SOURCE naive div 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 Metric inverse00 ORIGINAL rational div，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Metric inverse00 SOURCE naive div 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 — 以精確有理數重算 Metric determinant signed ORIGINAL rational sub，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Metric determinant signed SOURCE naive sub ORIGINAL numerator over denominator，不以小數近似替代。
• S023 · 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。
• S022 · premise_and_result_audit — 以精確有理數重算 Metric determinant cross ORIGINAL rational mul，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Metric determinant cross SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 Metric determinant main ORIGINAL rational mul，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Metric determinant main SOURCE naive mul 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 metric ORIGINAL matrix row1 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 metric ORIGINAL matrix row1 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 metric ORIGINAL matrix row0 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 metric ORIGINAL matrix row0 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 metric ORIGINAL covector 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 metric ORIGINAL covector 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

(non_euclidean_metric, [7, 1], [[2, 0], [0, 3]], [7/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)

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。