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R0028-14503 · q-20261014-advance-20261029-r03-inner-product-induced-norm-0053
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[4, 2], [2, 5]] and x=[-1, 0]. 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, 10]. 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=[[4, 2], [2, 5]] and x=[-1, 0]. 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, 10]. Give (inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass?,Cauchy_Schwarz_universal_certificate).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(4, 5)
After
inner parent diagonal product = 20
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(2, 2)
After
inner parent off-diagonal product = 4
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(20, 4)
After
inner parent determinant = 16
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=[[4, 2], [2, 5]]
After
symmetric=Yes;leading=4;determinant=16
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(2, 4)
After
inner parent completed-square slope = 1/2
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(16, 4)
After
inner parent completed-square remainder = 4
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=4;slope=1/2;positive remainder=4
After
q(s,t)=(4)*(s+(1/2)*t)^2+(4)*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
x self-inner-product:same matrix=[[4, 2], [2, 5]];first vector=[-1, 0];second vector=[-1, 0]
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, S010
Before
x self-inner-product matrix term0,0 := mul(4, -1)
After
x self-inner-product matrix term0,0 = -4
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
x self-inner-product matrix term0,1 := mul(2, 0)
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
x self-inner-product image coordinate0 := sum(-4, 0)
After
x self-inner-product image coordinate0 = -4
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
x self-inner-product matrix term1,0 := mul(2, -1)
After
x self-inner-product matrix term1,0 = -2
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
x self-inner-product matrix term1,1 := mul(5, 0)
After
x self-inner-product matrix term1,1 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
x self-inner-product image coordinate1 := sum(-2, 0)
After
x self-inner-product image coordinate1 = -2
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
x self-inner-product first-slot term0 := mul(-1, -4)
After
x self-inner-product first-slot term0 = 4
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
x self-inner-product first-slot term1 := mul(0, -2)
After
x self-inner-product first-slot term1 = 0
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
x self-inner-product inner sum := sum(4, 0)
After
x self-inner-product inner sum = 4
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
y self-inner-product:same matrix=[[4, 2], [2, 5]];first vector=[-3, 10];second vector=[-3, 10]
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, S020
Before
y self-inner-product matrix term0,0 := mul(4, -3)
After
y self-inner-product matrix term0,0 = -12
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
y self-inner-product matrix term0,1 := mul(2, 10)
After
y self-inner-product matrix term0,1 = 20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
y self-inner-product image coordinate0 := sum(-12, 20)
After
y self-inner-product image coordinate0 = 8
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
y self-inner-product matrix term1,0 := mul(2, -3)
After
y self-inner-product matrix term1,0 = -6
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
y self-inner-product matrix term1,1 := mul(5, 10)
After
y self-inner-product matrix term1,1 = 50
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
y self-inner-product image coordinate1 := sum(-6, 50)
After
y self-inner-product image coordinate1 = 44
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
y self-inner-product first-slot term0 := mul(-3, 8)
After
y self-inner-product first-slot term0 = -24
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
y self-inner-product first-slot term1 := mul(10, 44)
After
y self-inner-product first-slot term1 = 440
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
y self-inner-product inner sum := sum(-24, 440)
After
y self-inner-product inner sum = 416
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
ordered cross inner-product:same matrix=[[4, 2], [2, 5]];first vector=[-1, 0];second vector=[-3, 10]
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, S030
Before
ordered cross inner-product matrix term0,0 := mul(4, -3)
After
ordered cross inner-product matrix term0,0 = -12
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
ordered cross inner-product matrix term0,1 := mul(2, 10)
After
ordered cross inner-product matrix term0,1 = 20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
ordered cross inner-product image coordinate0 := sum(-12, 20)
After
ordered cross inner-product image coordinate0 = 8
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
ordered cross inner-product matrix term1,0 := mul(2, -3)
After
ordered cross inner-product matrix term1,0 = -6
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
ordered cross inner-product matrix term1,1 := mul(5, 10)
After
ordered cross inner-product matrix term1,1 = 50
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
ordered cross inner-product image coordinate1 := sum(-6, 50)
After
ordered cross inner-product image coordinate1 = 44
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
ordered cross inner-product first-slot term0 := mul(-1, 8)
After
ordered cross inner-product first-slot term0 = -8
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
ordered cross inner-product first-slot term1 := mul(0, 44)
After
ordered cross inner-product first-slot term1 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
ordered cross inner-product inner sum := sum(-8, 0)
After
ordered cross inner-product inner sum = -8
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
Cauchy-Schwarz signed inner squared := mul(-8, -8)
After
Cauchy-Schwarz signed inner squared = 64
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
Cauchy-Schwarz product of self-inner-products := mul(4, 416)
After
Cauchy-Schwarz product of self-inner-products = 1664
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
Cauchy-Schwarz projection scalar := div(-8, 416)
After
Cauchy-Schwarz projection scalar = -1/52
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
projection scaled y coordinate0 := mul(-1/52, -3)
After
projection scaled y coordinate0 = 3/52
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
projection residual coordinate0 := sub(-1, 3/52)
After
projection residual coordinate0 = -55/52
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
projection scaled y coordinate1 := mul(-1/52, 10)
After
projection scaled y coordinate1 = -5/26
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
projection residual coordinate1 := sub(0, -5/26)
After
projection residual coordinate1 = 5/26
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
projection residual self-inner-product:same matrix=[[4, 2], [2, 5]];first vector=[-55/52, 5/26];second vector=[-55/52, 5/26]
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, S047
Before
projection residual self-inner-product matrix term0,0 := mul(4, -55/52)
After
projection residual self-inner-product matrix term0,0 = -55/13
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
projection residual self-inner-product matrix term0,1 := mul(2, 5/26)
After
projection residual self-inner-product matrix term0,1 = 5/13
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
projection residual self-inner-product image coordinate0 := sum(-55/13, 5/13)
After
projection residual self-inner-product image coordinate0 = -50/13
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
projection residual self-inner-product matrix term1,0 := mul(2, -55/52)
After
projection residual self-inner-product matrix term1,0 = -55/26
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
projection residual self-inner-product matrix term1,1 := mul(5, 5/26)
After
projection residual self-inner-product matrix term1,1 = 25/26
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
projection residual self-inner-product image coordinate1 := sum(-55/26, 25/26)
After
projection residual self-inner-product image coordinate1 = -15/13
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
projection residual self-inner-product first-slot term0 := mul(-55/52, -50/13)
After
projection residual self-inner-product first-slot term0 = 1375/338
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
projection residual self-inner-product first-slot term1 := mul(5/26, -15/13)
After
projection residual self-inner-product first-slot term1 = -75/338
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
projection residual self-inner-product inner sum := sum(1375/338, -75/338)
After
projection residual self-inner-product inner sum = 50/13
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
yy=416>0;projection scalar=-1/52;residual=[-55/52, 5/26];residual square=50/13
After
q=xx-xy^2/yy>=0;positive multiplication gives xy^2<=xx*yy
Conditions: none
Evaluate the nonnegative residual square in the same form and expand by symmetry/bilinearity to xx−xy²/yy.Multiplication by positive yy preserves inequality;retain the original signed xy.
Rule: Cauchy_Schwarz_projection_identity_same_inner_product · Uses: G001, S057
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.(-8, 4, 416, 64, 1664, Yes, Cauchy_Schwarz_universal_certificate)
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: a8c96173dc53e78e595a45e0afda870e35c5848a69e28a735d3de42cd7570209candidate record SHA-256: b299a77ff75d1a2638c4ef37ba99a1ecb32e0cb00919ba13729854df2a8cf74esemantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209aequestion-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0Lean: NOT_RUN · human review: NOT_PERFORMED