Question 14509: A Cauchy–Schwarz Instance Retains the Universal-Certificate Boundary

Original question and givens

Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[10, 2], [2, 5]] and x=[-2, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[3, 4]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).

• G001 — Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[10, 2], [2, 5]] and x=[-2, 1]. The complete integer matrix must be symmetric with a positive leading entry and positive determinant. The induced norm is the nonnegative sqrt(<x,x>), with exact sqrt(...) text retained without simplification. Cauchy–Schwarz is a universal mathematical certificate, not proved by these vector instances or a new Lean receipt. Completeness and Hilbert structure are not inferred. y=[3, 4]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-inner-product-induced-norm

Step-by-step derivation

inner parent diagonal product := mul(10, 5)

inner parent diagonal product = 50

inner parent off-diagonal product := mul(2, 2)

inner parent off-diagonal product = 4

inner parent determinant := sub(50, 4)

inner parent determinant = 46

same real R2 inner-product candidate matrix=[[10, 2], [2, 5]]

symmetric=Yes;leading=10;determinant=46

inner parent completed-square slope := div(2, 10)

inner parent completed-square slope = 1/5

inner parent completed-square remainder := div(46, 10)

inner parent completed-square remainder = 23/5

positive leading=10;slope=1/5;positive remainder=23/5

q(s,t)=(10)*(s+(1/5)*t)^2+(23/5)*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=[[10, 2], [2, 5]];first vector=[-2, 1];second vector=[-2, 1]

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

x self-inner-product matrix term0,0 := mul(10, -2)

x self-inner-product matrix term0,0 = -20

x self-inner-product matrix term0,1 := mul(2, 1)

x self-inner-product matrix term0,1 = 2

x self-inner-product image coordinate0 := sum(-20, 2)

x self-inner-product image coordinate0 = -18

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

x self-inner-product matrix term1,0 = -4

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

x self-inner-product matrix term1,1 = 5

x self-inner-product image coordinate1 := sum(-4, 5)

x self-inner-product image coordinate1 = 1

x self-inner-product first-slot term0 := mul(-2, -18)

x self-inner-product first-slot term0 = 36

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(36, 1)

x self-inner-product inner sum = 37

y self-inner-product:same matrix=[[10, 2], [2, 5]];first vector=[3, 4];second vector=[3, 4]

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

y self-inner-product matrix term0,0 := mul(10, 3)

y self-inner-product matrix term0,0 = 30

y self-inner-product matrix term0,1 := mul(2, 4)

y self-inner-product matrix term0,1 = 8

y self-inner-product image coordinate0 := sum(30, 8)

y self-inner-product image coordinate0 = 38

y self-inner-product matrix term1,0 := mul(2, 3)

y self-inner-product matrix term1,0 = 6

y self-inner-product matrix term1,1 := mul(5, 4)

y self-inner-product matrix term1,1 = 20

y self-inner-product image coordinate1 := sum(6, 20)

y self-inner-product image coordinate1 = 26

y self-inner-product first-slot term0 := mul(3, 38)

y self-inner-product first-slot term0 = 114

y self-inner-product first-slot term1 := mul(4, 26)

y self-inner-product first-slot term1 = 104

y self-inner-product inner sum := sum(114, 104)

y self-inner-product inner sum = 218

ordered cross inner-product:same matrix=[[10, 2], [2, 5]];first vector=[-2, 1];second vector=[3, 4]

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

ordered cross inner-product matrix term0,0 := mul(10, 3)

ordered cross inner-product matrix term0,0 = 30

ordered cross inner-product matrix term0,1 := mul(2, 4)

ordered cross inner-product matrix term0,1 = 8

ordered cross inner-product image coordinate0 := sum(30, 8)

ordered cross inner-product image coordinate0 = 38

ordered cross inner-product matrix term1,0 := mul(2, 3)

ordered cross inner-product matrix term1,0 = 6

ordered cross inner-product matrix term1,1 := mul(5, 4)

ordered cross inner-product matrix term1,1 = 20

ordered cross inner-product image coordinate1 := sum(6, 20)

ordered cross inner-product image coordinate1 = 26

ordered cross inner-product first-slot term0 := mul(-2, 38)

ordered cross inner-product first-slot term0 = -76

ordered cross inner-product first-slot term1 := mul(1, 26)

ordered cross inner-product first-slot term1 = 26

ordered cross inner-product inner sum := sum(-76, 26)

ordered cross inner-product inner sum = -50

Cauchy-Schwarz signed inner squared := mul(-50, -50)

Cauchy-Schwarz signed inner squared = 2500

Cauchy-Schwarz product of self-inner-products := mul(37, 218)

Cauchy-Schwarz product of self-inner-products = 8066

Cauchy-Schwarz projection scalar := div(-50, 218)

Cauchy-Schwarz projection scalar = -25/109

projection scaled y coordinate0 := mul(-25/109, 3)

projection scaled y coordinate0 = -75/109

projection residual coordinate0 := sub(-2, -75/109)

projection residual coordinate0 = -143/109

projection scaled y coordinate1 := mul(-25/109, 4)

projection scaled y coordinate1 = -100/109

projection residual coordinate1 := sub(1, -100/109)

projection residual coordinate1 = 209/109

projection residual self-inner-product:same matrix=[[10, 2], [2, 5]];first vector=[-143/109, 209/109];second vector=[-143/109, 209/109]

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

projection residual self-inner-product matrix term0,0 := mul(10, -143/109)

projection residual self-inner-product matrix term0,0 = -1430/109

projection residual self-inner-product matrix term0,1 := mul(2, 209/109)

projection residual self-inner-product matrix term0,1 = 418/109

projection residual self-inner-product image coordinate0 := sum(-1430/109, 418/109)

projection residual self-inner-product image coordinate0 = -1012/109

projection residual self-inner-product matrix term1,0 := mul(2, -143/109)

projection residual self-inner-product matrix term1,0 = -286/109

projection residual self-inner-product matrix term1,1 := mul(5, 209/109)

projection residual self-inner-product matrix term1,1 = 1045/109

projection residual self-inner-product image coordinate1 := sum(-286/109, 1045/109)

projection residual self-inner-product image coordinate1 = 759/109

projection residual self-inner-product first-slot term0 := mul(-143/109, -1012/109)

projection residual self-inner-product first-slot term0 = 144716/11881

projection residual self-inner-product first-slot term1 := mul(209/109, 759/109)

projection residual self-inner-product first-slot term1 = 158631/11881

projection residual self-inner-product inner sum := sum(144716/11881, 158631/11881)

projection residual self-inner-product inner sum = 2783/109

yy=218>0;projection scalar=-25/109;residual=[-143/109, 209/109];residual square=2783/109

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

Reverse checks

• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Recompute projection residual self-inner-product inner sum with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute projection residual self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute projection residual self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute projection residual self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute projection residual self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute projection residual self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute projection residual self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute projection residual self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute projection residual self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Recompute projection residual coordinate1 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute projection scaled y coordinate1 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute projection residual coordinate0 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute projection scaled y coordinate0 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Cauchy-Schwarz projection scalar with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Cauchy-Schwarz product of self-inner-products with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Cauchy-Schwarz signed inner squared with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute ordered cross inner-product inner sum with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute ordered cross inner-product first-slot term1 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute ordered cross inner-product first-slot term0 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute ordered cross inner-product image coordinate1 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute ordered cross inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute ordered cross inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute ordered cross inner-product image coordinate0 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute ordered cross inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute ordered cross inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute y self-inner-product inner sum with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute y self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute y self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute y self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute y self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute y self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute y self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute y self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute y self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute x self-inner-product inner sum with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute x self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute x self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute x self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute x self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute x self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute x self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute x self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute x self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute inner parent completed-square remainder with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute inner parent completed-square slope with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute inner parent determinant with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute inner parent off-diagonal product with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute inner parent diagonal product with exact rational arithmetic, not rounded decimals.

Result

(-50, 37, 218, 2500, 8066, Yes, Cauchy_Schwarz_universal_certificate)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.