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R0028-14545 · q-20261014-advance-20261029-r03-inner-product-induced-norm-0045
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[21, 1], [1, 2]] and x=[2, 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=[3, -7]. 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=[[21, 1], [1, 2]] and x=[2, 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=[3, -7]. 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(21, 2)
After
inner parent diagonal product = 42
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(1, 1)
After
inner parent off-diagonal product = 1
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(42, 1)
After
inner parent determinant = 41
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=[[21, 1], [1, 2]]
After
symmetric=Yes;leading=21;determinant=41
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(1, 21)
After
inner parent completed-square slope = 1/21
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(41, 21)
After
inner parent completed-square remainder = 41/21
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=21;slope=1/21;positive remainder=41/21
After
q(s,t)=(21)*(s+(1/21)*t)^2+(41/21)*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(2, 3)
After
distance signed coordinate0 = -1
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, -7)
After
distance signed coordinate1 = 9
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=[[21, 1], [1, 2]];first vector=[-1, 9];second vector=[-1, 9]
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(21, -1)
After
distance difference square matrix term0,0 = -21
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(1, 9)
After
distance difference square matrix term0,1 = 9
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(-21, 9)
After
distance difference square image coordinate0 = -12
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(1, -1)
After
distance difference square matrix term1,0 = -1
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(2, 9)
After
distance difference square matrix term1,1 = 18
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(-1, 18)
After
distance difference square image coordinate1 = 17
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(-1, -12)
After
distance difference square first-slot term0 = 12
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(9, 17)
After
distance difference square first-slot term1 = 153
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(12, 153)
After
distance difference square inner sum = 165
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=165>=0
After
nonnegative exact root=sqrt(165);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.([2, 2], [3, -7], [-1, 9], 165, sqrt(165))
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: 815f2a2c12f629a7aa46e64222ee993d1263e511a16f53a99bdaadf14e55c72fcandidate record SHA-256: a01eadb11c7729fb127733b1af5eeed330b2b892c4d4122fac077227078811f4semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209aequestion-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0Lean: NOT_RUN · human review: NOT_PERFORMED