第 14508 題：內積誘導範數

原題與已知條件

向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={"matrix":[[9,1],[1,2]],"x":[-3,0],"y":[2,5]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。

• G001 — 向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={"matrix":[[9,1],[1,2]],"x":[-3,0],"y":[2,5]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。

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-inner-product-induced-norm

逐步推導

inner parent diagonal product := mul(9, 2)

inner parent diagonal product = 18

inner parent off-diagonal product := mul(1, 1)

inner parent off-diagonal product = 1

inner parent determinant := sub(18, 1)

inner parent determinant = 17

same real R2 inner-product candidate matrix=[[9, 1], [1, 2]]

symmetric=是;leading=9;determinant=17

inner parent completed-square slope := div(1, 9)

inner parent completed-square slope = 1/9

inner parent completed-square remainder := div(17, 9)

inner parent completed-square remainder = 17/9

positive leading=9;slope=1/9;positive remainder=17/9

q(s,t)=(9)*(s+(1/9)*t)^2+(17/9)*t^2;positive iff(s,t)!=0

same certified real inner product;arbitrary real x,y

if y=0 then inner(x,y)=0;otherwise yy>0 and q(x-a*y)=xx-2*a*xy+a^2*yy>=0

norm(x)=nonnegative sqrt(inner(x,x));inner product positive definite and bilinear

norm(x)>=0;norm(x)=0 iff x=0;norm(a*x)=abs(a)*norm(x)

Cauchy-Schwarz holds for arbitrary x,y in this same inner product

norm(x+y)^2=xx+2*xy+yy<=(norm(x)+norm(y))^2;nonnegative roots give triangle inequality

x self-inner-product:same matrix=[[9, 1], [1, 2]];first vector=[-3, 0];second vector=[-3, 0]

inner product=first^T*(matrix*second)

x self-inner-product matrix term0,0 := mul(9, -3)

x self-inner-product matrix term0,0 = -27

x self-inner-product matrix term0,1 := mul(1, 0)

x self-inner-product matrix term0,1 = 0

x self-inner-product image coordinate0 := sum(-27, 0)

x self-inner-product image coordinate0 = -27

x self-inner-product matrix term1,0 := mul(1, -3)

x self-inner-product matrix term1,0 = -3

x self-inner-product matrix term1,1 := mul(2, 0)

x self-inner-product matrix term1,1 = 0

x self-inner-product image coordinate1 := sum(-3, 0)

x self-inner-product image coordinate1 = -3

x self-inner-product first-slot term0 := mul(-3, -27)

x self-inner-product first-slot term0 = 81

x self-inner-product first-slot term1 := mul(0, -3)

x self-inner-product first-slot term1 = 0

x self-inner-product inner sum := sum(81, 0)

x self-inner-product inner sum = 81

y self-inner-product:same matrix=[[9, 1], [1, 2]];first vector=[2, 5];second vector=[2, 5]

inner product=first^T*(matrix*second)

y self-inner-product matrix term0,0 := mul(9, 2)

y self-inner-product matrix term0,0 = 18

y self-inner-product matrix term0,1 := mul(1, 5)

y self-inner-product matrix term0,1 = 5

y self-inner-product image coordinate0 := sum(18, 5)

y self-inner-product image coordinate0 = 23

y self-inner-product matrix term1,0 := mul(1, 2)

y self-inner-product matrix term1,0 = 2

y self-inner-product matrix term1,1 := mul(2, 5)

y self-inner-product matrix term1,1 = 10

y self-inner-product image coordinate1 := sum(2, 10)

y self-inner-product image coordinate1 = 12

y self-inner-product first-slot term0 := mul(2, 23)

y self-inner-product first-slot term0 = 46

y self-inner-product first-slot term1 := mul(5, 12)

y self-inner-product first-slot term1 = 60

y self-inner-product inner sum := sum(46, 60)

y self-inner-product inner sum = 106

ordered cross inner-product:same matrix=[[9, 1], [1, 2]];first vector=[-3, 0];second vector=[2, 5]

inner product=first^T*(matrix*second)

ordered cross inner-product matrix term0,0 := mul(9, 2)

ordered cross inner-product matrix term0,0 = 18

ordered cross inner-product matrix term0,1 := mul(1, 5)

ordered cross inner-product matrix term0,1 = 5

ordered cross inner-product image coordinate0 := sum(18, 5)

ordered cross inner-product image coordinate0 = 23

ordered cross inner-product matrix term1,0 := mul(1, 2)

ordered cross inner-product matrix term1,0 = 2

ordered cross inner-product matrix term1,1 := mul(2, 5)

ordered cross inner-product matrix term1,1 = 10

ordered cross inner-product image coordinate1 := sum(2, 10)

ordered cross inner-product image coordinate1 = 12

ordered cross inner-product first-slot term0 := mul(-3, 23)

ordered cross inner-product first-slot term0 = -69

ordered cross inner-product first-slot term1 := mul(0, 12)

ordered cross inner-product first-slot term1 = 0

ordered cross inner-product inner sum := sum(-69, 0)

ordered cross inner-product inner sum = -69

Cauchy-Schwarz signed inner squared := mul(-69, -69)

Cauchy-Schwarz signed inner squared = 4761

Cauchy-Schwarz product of self-inner-products := mul(81, 106)

Cauchy-Schwarz product of self-inner-products = 8586

Cauchy-Schwarz projection scalar := div(-69, 106)

Cauchy-Schwarz projection scalar = -69/106

projection scaled y coordinate0 := mul(-69/106, 2)

projection scaled y coordinate0 = -69/53

projection residual coordinate0 := sub(-3, -69/53)

projection residual coordinate0 = -90/53

projection scaled y coordinate1 := mul(-69/106, 5)

projection scaled y coordinate1 = -345/106

projection residual coordinate1 := sub(0, -345/106)

projection residual coordinate1 = 345/106

projection residual self-inner-product:same matrix=[[9, 1], [1, 2]];first vector=[-90/53, 345/106];second vector=[-90/53, 345/106]

inner product=first^T*(matrix*second)

projection residual self-inner-product matrix term0,0 := mul(9, -90/53)

projection residual self-inner-product matrix term0,0 = -810/53

projection residual self-inner-product matrix term0,1 := mul(1, 345/106)

projection residual self-inner-product matrix term0,1 = 345/106

projection residual self-inner-product image coordinate0 := sum(-810/53, 345/106)

projection residual self-inner-product image coordinate0 = -1275/106

projection residual self-inner-product matrix term1,0 := mul(1, -90/53)

projection residual self-inner-product matrix term1,0 = -90/53

projection residual self-inner-product matrix term1,1 := mul(2, 345/106)

projection residual self-inner-product matrix term1,1 = 345/53

projection residual self-inner-product image coordinate1 := sum(-90/53, 345/53)

projection residual self-inner-product image coordinate1 = 255/53

projection residual self-inner-product first-slot term0 := mul(-90/53, -1275/106)

projection residual self-inner-product first-slot term0 = 57375/2809

projection residual self-inner-product first-slot term1 := mul(345/106, 255/53)

projection residual self-inner-product first-slot term1 = 87975/5618

projection residual self-inner-product inner sum := sum(57375/2809, 87975/5618)

projection residual self-inner-product inner sum = 3825/106

yy=106>0;projection scalar=-69/106;residual=[-90/53, 345/106];residual square=3825/106

q=xx-xy^2/yy>=0;positive multiplication gives xy^2<=xx*yy

逐步反向核對

• S058 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S057 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product inner sum，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product first-slot term1，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product first-slot term0，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product image coordinate1，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product matrix term1,1，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product matrix term1,0，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product image coordinate0，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product matrix term0,1，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product matrix term0,0，不以小數近似替代。
• S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S047 · premise_and_result_audit — 以精確有理數重算 projection residual coordinate1，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 projection scaled y coordinate1，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 projection residual coordinate0，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 projection scaled y coordinate0，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Cauchy-Schwarz projection scalar，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Cauchy-Schwarz product of self-inner-products，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Cauchy-Schwarz signed inner squared，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product inner sum，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product first-slot term1，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product first-slot term0，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product image coordinate1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product matrix term1,1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product matrix term1,0，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product image coordinate0，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product matrix term0,1，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product matrix term0,0，不以小數近似替代。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 以精確有理數重算 y self-inner-product inner sum，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 y self-inner-product first-slot term1，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 y self-inner-product first-slot term0，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 y self-inner-product image coordinate1，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 y self-inner-product matrix term1,1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 y self-inner-product matrix term1,0，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 y self-inner-product image coordinate0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 y self-inner-product matrix term0,1，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 y self-inner-product matrix term0,0，不以小數近似替代。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 以精確有理數重算 x self-inner-product inner sum，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 x self-inner-product first-slot term1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 x self-inner-product first-slot term0，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 x self-inner-product image coordinate1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 x self-inner-product matrix term1,1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 x self-inner-product matrix term1,0，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 x self-inner-product image coordinate0，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 x self-inner-product matrix term0,1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 x self-inner-product matrix term0,0，不以小數近似替代。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 inner parent completed-square remainder，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 inner parent completed-square slope，不以小數近似替代。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 以精確有理數重算 inner parent determinant，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 inner parent off-diagonal product，不以小數近似替代。
• S001 · premise_and_result_audit — 以精確有理數重算 inner parent diagonal product，不以小數近似替代。

結論

(-69, 81, 106, 4761, 8586, 是, Cauchy_Schwarz_universal_certificate)

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。