# Question 14582: Use the Nonnegative Exact Root of the Same Self-Inner-Product

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0028-14582 · q-20261014-advance-20261029-r03-inner-product-induced-norm-0007



## Original question and givens

Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[8, 0], [0, 1]] and x=[3, -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. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_certificate).


- G001 — Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[8, 0], [0, 1]] and x=[3, -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. Give (matrix,x,norm_square,exact_norm,same_inner_product_universal_norm_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](https://ocme.evemisslab.com/en/mko/mko-inner-product-induced-norm/)





## Step-by-step derivation

S001 · calculationBefore

inner parent diagonal product := mul(8, 1)

After

inner parent diagonal product = 8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · calculationBefore

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

After

inner parent off-diagonal product = 0

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

inner parent determinant := sub(8, 0)

After

inner parent determinant = 8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

same real R2 inner-product candidate matrix=[[8, 0], [0, 1]]

After

symmetric=Yes;leading=8;determinant=8

Conditions: none

Qualify the same symmetric positive-definite real R² parent matrix,not an unrelated Euclidean or coordinate norm.Symmetry,positive leading entry and positive determinant are required.

Step identifiers and dependenciesRule: innernorm_certified_same_SPD_parent · Uses: G001, S003





S005 · calculationBefore

inner parent completed-square slope := div(0, 8)

After

inner parent completed-square slope = 0

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

inner parent completed-square remainder := div(8, 8)

After

inner parent completed-square remainder = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

positive leading=8;slope=0;positive remainder=1

After

q(s,t)=(8)*(s+(0)*t)^2+(1)*t^2;positive iff(s,t)!=0

Conditions: none

The two positive completed-square coefficients give positive self-inner-product for every nonzero real vector and0 at zero,not merely sampled positive values.

Step identifiers and dependenciesRule: innernorm_parent_global_positive_square_certificate · Uses: G001, S006





S008 · impliesBefore

same certified real inner product;arbitrary real x,y

After

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

Conditions: none

For every real a,self-inner-product is nonnegative.When y≠0,yy>0;choose a=xy/yy to obtain xx−xy²/yy≥0,multiply by positive yy to get xy²≤xx*yy.Handle y=0 separately without0/0.

Step identifiers and dependenciesRule: innernorm_Cauchy_Schwarz_all_vectors_quadratic · Uses: G001, S007





S009 · impliesBefore

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

After

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

Conditions: none

Only the nonnegative root is the norm.From q(ax)=a²q(x),taking roots gives|a|,not a,including negative scalars.The zero and homogeneity claims are universal.

Step identifiers and dependenciesRule: innernorm_nonnegative_root_and_absolute_homogeneity · Uses: G001, S008





S010 · impliesBefore

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

After

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

Conditions: none

Bound the cross term by|xy|≤norm(x)norm(y),then take nonnegative roots for the universal triangle inequality.One exact instance is not the theorem's source,and no completeness follows.

Step identifiers and dependenciesRule: innernorm_triangle_uses_universal_Cauchy_Schwarz · Uses: G001, S009





S011 · impliesBefore

x self-inner-product:same matrix=[[8, 0], [0, 1]];first vector=[3, -1];second vector=[3, -1]

After

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

Conditions: none

Use the same parent inner product:matrix times second slot,then dot with first.Derived rational vectors still belong to R² and are not clipped to raw integer input bounds.

Step identifiers and dependenciesRule: innernorm_original_slot_roles · Uses: G001, S010





S012 · calculationBefore

x self-inner-product matrix term0,0 := mul(8, 3)

After

x self-inner-product matrix term0,0 = 24

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

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

After

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

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

x self-inner-product image coordinate0 := sum(24, 0)

After

x self-inner-product image coordinate0 = 24

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

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

After

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

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

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

After

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

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

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

After

x self-inner-product image coordinate1 = -1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

x self-inner-product first-slot term0 := mul(3, 24)

After

x self-inner-product first-slot term0 = 72

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

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

After

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

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

x self-inner-product inner sum := sum(72, 1)

After

x self-inner-product inner sum = 73

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · impliesBefore

induced norm:same-form self-inner-product=73>=0

After

nonnegative exact root=sqrt(73);source expression kept unsimplified

Conditions: none

Retain the exact source sqrt(...) string even for a perfect square,without integer/decimal replacement.The negative root is not a norm/distance,and no other inner product or norm is substituted.

Step identifiers and dependenciesRule: innernorm_exact_root_not_decimal_or_other_metric · Uses: G001, S020









## Reverse checks


- 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

([[8, 0], [0, 1]], [3, -1], 73, sqrt(73), same_inner_product_universal_norm_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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 81d05f8f23e1386f92cc27a25102f018e5073caaea3b3bbad30e40a8447aeb35
- candidate record SHA-256: 5006c6439d8eb67520d6457c49f33191dbe0a951a6e0f75a3902a6b9a20eebed
- semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae
- question-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0
- Lean: NOT_RUN · human review: NOT_PERFORMED

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