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R0028-14517 · q-20261014-advance-20261029-r03-inner-product-induced-norm-0067
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[18, 1], [1, 2]] and x=[-1, -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=[0, -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=[[18, 1], [1, 2]] and x=[-1, -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=[0, -4]. 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(18, 2)
After
inner parent diagonal product = 36
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(36, 1)
After
inner parent determinant = 35
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=[[18, 1], [1, 2]]
After
symmetric=Yes;leading=18;determinant=35
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, 18)
After
inner parent completed-square slope = 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, S004
Before
inner parent completed-square remainder := div(35, 18)
After
inner parent completed-square remainder = 35/18
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=18;slope=1/18;positive remainder=35/18
After
q(s,t)=(18)*(s+(1/18)*t)^2+(35/18)*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=[[18, 1], [1, 2]];first vector=[-1, -1];second vector=[-1, -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, S010
Before
x self-inner-product matrix term0,0 := mul(18, -1)
After
x self-inner-product matrix term0,0 = -18
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(1, -1)
After
x self-inner-product matrix term0,1 = -1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
x self-inner-product image coordinate0 := sum(-18, -1)
After
x self-inner-product image coordinate0 = -19
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(1, -1)
After
x self-inner-product 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, S014
Before
x self-inner-product matrix term1,1 := mul(2, -1)
After
x self-inner-product matrix term1,1 = -2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
x self-inner-product image coordinate1 := sum(-1, -2)
After
x self-inner-product image coordinate1 = -3
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, -19)
After
x self-inner-product first-slot term0 = 19
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(-1, -3)
After
x self-inner-product first-slot term1 = 3
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(19, 3)
After
x self-inner-product inner sum = 22
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=[[18, 1], [1, 2]];first vector=[0, -4];second vector=[0, -4]
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(18, 0)
After
y self-inner-product matrix term0,0 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
y self-inner-product matrix term0,1 := mul(1, -4)
After
y self-inner-product matrix term0,1 = -4
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(0, -4)
After
y 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, S023
Before
y self-inner-product matrix term1,0 := mul(1, 0)
After
y self-inner-product 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, S024
Before
y self-inner-product matrix term1,1 := mul(2, -4)
After
y self-inner-product matrix term1,1 = -8
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(0, -8)
After
y self-inner-product image coordinate1 = -8
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(0, -4)
After
y self-inner-product first-slot term0 = 0
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(-4, -8)
After
y self-inner-product first-slot term1 = 32
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(0, 32)
After
y self-inner-product inner sum = 32
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=[[18, 1], [1, 2]];first vector=[-1, -1];second vector=[0, -4]
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(18, 0)
After
ordered cross inner-product matrix term0,0 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
ordered cross inner-product matrix term0,1 := mul(1, -4)
After
ordered cross inner-product matrix term0,1 = -4
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(0, -4)
After
ordered cross 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, S033
Before
ordered cross inner-product matrix term1,0 := mul(1, 0)
After
ordered cross inner-product 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, S034
Before
ordered cross inner-product matrix term1,1 := mul(2, -4)
After
ordered cross inner-product matrix term1,1 = -8
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(0, -8)
After
ordered cross inner-product image coordinate1 = -8
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, -4)
After
ordered cross 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, S037
Before
ordered cross inner-product first-slot term1 := mul(-1, -8)
After
ordered cross inner-product first-slot term1 = 8
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(4, 8)
After
ordered cross inner-product inner sum = 12
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(12, 12)
After
Cauchy-Schwarz signed inner squared = 144
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(22, 32)
After
Cauchy-Schwarz product of self-inner-products = 704
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(12, 32)
After
Cauchy-Schwarz projection scalar = 3/8
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(3/8, 0)
After
projection scaled y coordinate0 = 0
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, 0)
After
projection residual 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, S044
Before
projection scaled y coordinate1 := mul(3/8, -4)
After
projection scaled y coordinate1 = -3/2
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(-1, -3/2)
After
projection residual coordinate1 = 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, S046
Before
projection residual self-inner-product:same matrix=[[18, 1], [1, 2]];first vector=[-1, 1/2];second vector=[-1, 1/2]
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(18, -1)
After
projection residual self-inner-product matrix term0,0 = -18
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(1, 1/2)
After
projection residual self-inner-product matrix term0,1 = 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, S049
Before
projection residual self-inner-product image coordinate0 := sum(-18, 1/2)
After
projection residual self-inner-product image coordinate0 = -35/2
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(1, -1)
After
projection residual self-inner-product 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, S051
Before
projection residual self-inner-product matrix term1,1 := mul(2, 1/2)
After
projection residual self-inner-product 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, S052
Before
projection residual self-inner-product image coordinate1 := sum(-1, 1)
After
projection residual self-inner-product image coordinate1 = 0
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(-1, -35/2)
After
projection residual self-inner-product first-slot term0 = 35/2
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(1/2, 0)
After
projection residual 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, S055
Before
projection residual self-inner-product inner sum := sum(35/2, 0)
After
projection residual self-inner-product inner sum = 35/2
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=32>0;projection scalar=3/8;residual=[-1, 1/2];residual square=35/2
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.(12, 22, 32, 144, 704, 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: 4151004cef2fbf50a91eb7ee95afd8aca870510ac1e4ab9f0a4edd2a316808f8candidate record SHA-256: 81fe79cbe2f344e674cb20b7806c6ee74f5156c405d9b2303e580306112e9815semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209aequestion-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0Lean: NOT_RUN · human review: NOT_PERFORMED