第 14522 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":[[23,0],[0,1]],"x":[-3,-1],"y":[5,-9]}。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":[[23,0],[0,1]],"x":[-3,-1],"y":[5,-9]}。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(23, 1)

inner parent diagonal product = 23

inner parent off-diagonal product := mul(0, 0)

inner parent off-diagonal product = 0

inner parent determinant := sub(23, 0)

inner parent determinant = 23

same real R2 inner-product candidate matrix=[[23, 0], [0, 1]]

symmetric=是;leading=23;determinant=23

inner parent completed-square slope := div(0, 23)

inner parent completed-square slope = 0

inner parent completed-square remainder := div(23, 23)

inner parent completed-square remainder = 1

positive leading=23;slope=0;positive remainder=1

q(s,t)=(23)*(s+(0)*t)^2+(1)*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=[[23, 0], [0, 1]];first vector=[-3, -1];second vector=[-3, -1]

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

x self-inner-product matrix term0,0 := mul(23, -3)

x self-inner-product matrix term0,0 = -69

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

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

x self-inner-product image coordinate0 := sum(-69, 0)

x self-inner-product image coordinate0 = -69

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

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

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

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

x self-inner-product image coordinate1 := sum(0, -1)

x self-inner-product image coordinate1 = -1

x self-inner-product first-slot term0 := mul(-3, -69)

x self-inner-product first-slot term0 = 207

x self-inner-product first-slot term1 := mul(-1, -1)

x self-inner-product first-slot term1 = 1

x self-inner-product inner sum := sum(207, 1)

x self-inner-product inner sum = 208

y self-inner-product:same matrix=[[23, 0], [0, 1]];first vector=[5, -9];second vector=[5, -9]

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

y self-inner-product matrix term0,0 := mul(23, 5)

y self-inner-product matrix term0,0 = 115

y self-inner-product matrix term0,1 := mul(0, -9)

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

y self-inner-product image coordinate0 := sum(115, 0)

y self-inner-product image coordinate0 = 115

y self-inner-product matrix term1,0 := mul(0, 5)

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

y self-inner-product matrix term1,1 := mul(1, -9)

y self-inner-product matrix term1,1 = -9

y self-inner-product image coordinate1 := sum(0, -9)

y self-inner-product image coordinate1 = -9

y self-inner-product first-slot term0 := mul(5, 115)

y self-inner-product first-slot term0 = 575

y self-inner-product first-slot term1 := mul(-9, -9)

y self-inner-product first-slot term1 = 81

y self-inner-product inner sum := sum(575, 81)

y self-inner-product inner sum = 656

ordered cross inner-product:same matrix=[[23, 0], [0, 1]];first vector=[-3, -1];second vector=[5, -9]

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

ordered cross inner-product matrix term0,0 := mul(23, 5)

ordered cross inner-product matrix term0,0 = 115

ordered cross inner-product matrix term0,1 := mul(0, -9)

ordered cross inner-product matrix term0,1 = 0

ordered cross inner-product image coordinate0 := sum(115, 0)

ordered cross inner-product image coordinate0 = 115

ordered cross inner-product matrix term1,0 := mul(0, 5)

ordered cross inner-product matrix term1,0 = 0

ordered cross inner-product matrix term1,1 := mul(1, -9)

ordered cross inner-product matrix term1,1 = -9

ordered cross inner-product image coordinate1 := sum(0, -9)

ordered cross inner-product image coordinate1 = -9

ordered cross inner-product first-slot term0 := mul(-3, 115)

ordered cross inner-product first-slot term0 = -345

ordered cross inner-product first-slot term1 := mul(-1, -9)

ordered cross inner-product first-slot term1 = 9

ordered cross inner-product inner sum := sum(-345, 9)

ordered cross inner-product inner sum = -336

Cauchy-Schwarz signed inner squared := mul(-336, -336)

Cauchy-Schwarz signed inner squared = 112896

Cauchy-Schwarz product of self-inner-products := mul(208, 656)

Cauchy-Schwarz product of self-inner-products = 136448

Cauchy-Schwarz projection scalar := div(-336, 656)

Cauchy-Schwarz projection scalar = -21/41

projection scaled y coordinate0 := mul(-21/41, 5)

projection scaled y coordinate0 = -105/41

projection residual coordinate0 := sub(-3, -105/41)

projection residual coordinate0 = -18/41

projection scaled y coordinate1 := mul(-21/41, -9)

projection scaled y coordinate1 = 189/41

projection residual coordinate1 := sub(-1, 189/41)

projection residual coordinate1 = -230/41

projection residual self-inner-product:same matrix=[[23, 0], [0, 1]];first vector=[-18/41, -230/41];second vector=[-18/41, -230/41]

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

projection residual self-inner-product matrix term0,0 := mul(23, -18/41)

projection residual self-inner-product matrix term0,0 = -414/41

projection residual self-inner-product matrix term0,1 := mul(0, -230/41)

projection residual self-inner-product matrix term0,1 = 0

projection residual self-inner-product image coordinate0 := sum(-414/41, 0)

projection residual self-inner-product image coordinate0 = -414/41

projection residual self-inner-product matrix term1,0 := mul(0, -18/41)

projection residual self-inner-product matrix term1,0 = 0

projection residual self-inner-product matrix term1,1 := mul(1, -230/41)

projection residual self-inner-product matrix term1,1 = -230/41

projection residual self-inner-product image coordinate1 := sum(0, -230/41)

projection residual self-inner-product image coordinate1 = -230/41

projection residual self-inner-product first-slot term0 := mul(-18/41, -414/41)

projection residual self-inner-product first-slot term0 = 7452/1681

projection residual self-inner-product first-slot term1 := mul(-230/41, -230/41)

projection residual self-inner-product first-slot term1 = 52900/1681

projection residual self-inner-product inner sum := sum(7452/1681, 52900/1681)

projection residual self-inner-product inner sum = 1472/41

yy=656>0;projection scalar=-21/41;residual=[-18/41, -230/41];residual square=1472/41

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

(-336, 208, 656, 112896, 136448, 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。