Explore related knowledge and questions →
← Question derivation index · Original question batch
R0028-14535 · q-20261014-advance-20261029-r03-inner-product-induced-norm-0035
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[11, 0], [0, 1]] and x=[-1, 2]. 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=[4, 3]. Give (x,y,difference=x−y,distance_square,exact_distance).
G001 — Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[11, 0], [0, 1]] and x=[-1, 2]. 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=[4, 3]. Give (x,y,difference=x−y,distance_square,exact_distance).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.
Before
inner parent diagonal product := mul(11, 1)
After
inner parent diagonal product = 11
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
inner parent determinant := sub(11, 0)
After
inner parent determinant = 11
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
same real R2 inner-product candidate matrix=[[11, 0], [0, 1]]
After
symmetric=Yes;leading=11;determinant=11
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.
Rule: innernorm_certified_same_SPD_parent · Uses: G001, S003
Before
inner parent completed-square slope := div(0, 11)
After
inner parent completed-square slope = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
inner parent completed-square remainder := div(11, 11)
After
inner parent completed-square remainder = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
positive leading=11;slope=0;positive remainder=1
After
q(s,t)=(11)*(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.
Rule: innernorm_parent_global_positive_square_certificate · Uses: G001, S006
Before
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.
Rule: innernorm_Cauchy_Schwarz_all_vectors_quadratic · Uses: G001, S007
Before
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.
Rule: innernorm_nonnegative_root_and_absolute_homogeneity · Uses: G001, S008
Before
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.
Rule: innernorm_triangle_uses_universal_Cauchy_Schwarz · Uses: G001, S009
Before
distance signed coordinate0 := sub(-1, 4)
After
distance signed coordinate0 = -5
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
distance signed coordinate1 := sub(2, 3)
After
distance signed coordinate1 = -1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
distance difference square:same matrix=[[11, 0], [0, 1]];first vector=[-5, -1];second vector=[-5, -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.
Rule: innernorm_original_slot_roles · Uses: G001, S012
Before
distance difference square matrix term0,0 := mul(11, -5)
After
distance difference square matrix term0,0 = -55
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
distance difference square matrix term0,1 := mul(0, -1)
After
distance difference square matrix term0,1 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
distance difference square image coordinate0 := sum(-55, 0)
After
distance difference square image coordinate0 = -55
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
distance difference square matrix term1,0 := mul(0, -5)
After
distance difference square matrix term1,0 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
distance difference square matrix term1,1 := mul(1, -1)
After
distance difference square matrix term1,1 = -1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
distance difference square image coordinate1 := sum(0, -1)
After
distance difference square image coordinate1 = -1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
distance difference square first-slot term0 := mul(-5, -55)
After
distance difference square first-slot term0 = 275
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
distance difference square first-slot term1 := mul(-1, -1)
After
distance difference square first-slot term1 = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
distance difference square inner sum := sum(275, 1)
After
distance difference square inner sum = 276
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
induced distance:same-form self-inner-product=276>=0
After
nonnegative exact root=sqrt(276);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.
Rule: innernorm_exact_root_not_decimal_or_other_metric · Uses: G001, S022
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Recompute distance difference square inner sum with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute distance difference square first-slot term1 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute distance difference square first-slot term0 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute distance difference square image coordinate1 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute distance difference square matrix term1,1 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute distance difference square matrix term1,0 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute distance difference square image coordinate0 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute distance difference square matrix term0,1 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute distance difference square matrix term0,0 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Recompute distance signed coordinate1 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute distance signed coordinate0 with exact rational arithmetic, not rounded decimals.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.([-1, 2], [4, 3], [-5, -1], 276, sqrt(276))
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 47799801a0edc9727c3084521199db2642b074b6b0f313caf7dd70ddae432f50candidate record SHA-256: a98a8f830e973720513406b59de2b77433daf9758c63e9b5059158b9224a99c2semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209aequestion-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0Lean: NOT_RUN · human review: NOT_PERFORMED