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R0028-14559 · q-20261014-advance-20261029-r03-inner-product-induced-norm-0084
Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[10, 2], [2, 5]] and x=[-2, 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=[3, 4]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).
G001 — Use the same certified real inner product on R², <u,v>=u^T A v, with A=[[10, 2], [2, 5]] and x=[-2, 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=[3, 4]. Give (x,y,lhs=norm_square(x+y)+norm_square(x−y),rhs=2*norm_square(x)+2*norm_square(y),equal?).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(10, 5)
After
inner parent diagonal product = 50
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(50, 4)
After
inner parent determinant = 46
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=[[10, 2], [2, 5]]
After
symmetric=Yes;leading=10;determinant=46
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, 10)
After
inner parent completed-square slope = 1/5
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(46, 10)
After
inner parent completed-square remainder = 23/5
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=10;slope=1/5;positive remainder=23/5
After
q(s,t)=(10)*(s+(1/5)*t)^2+(23/5)*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=[[10, 2], [2, 5]];first vector=[-2, 1];second vector=[-2, 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(10, -2)
After
x self-inner-product matrix term0,0 = -20
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, 1)
After
x self-inner-product matrix term0,1 = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
x self-inner-product image coordinate0 := sum(-20, 2)
After
x self-inner-product image coordinate0 = -18
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, -2)
After
x self-inner-product matrix term1,0 = -4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
x self-inner-product matrix term1,1 := mul(5, 1)
After
x self-inner-product matrix term1,1 = 5
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(-4, 5)
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.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
x self-inner-product first-slot term0 := mul(-2, -18)
After
x self-inner-product first-slot term0 = 36
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, 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.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
x self-inner-product inner sum := sum(36, 1)
After
x self-inner-product inner sum = 37
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=[[10, 2], [2, 5]];first vector=[3, 4];second vector=[3, 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(10, 3)
After
y self-inner-product matrix term0,0 = 30
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, 4)
After
y self-inner-product matrix term0,1 = 8
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
y self-inner-product image coordinate0 := sum(30, 8)
After
y self-inner-product image coordinate0 = 38
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, 4)
After
y self-inner-product matrix term1,1 = 20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
y self-inner-product image coordinate1 := sum(6, 20)
After
y self-inner-product image coordinate1 = 26
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, 38)
After
y self-inner-product first-slot term0 = 114
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, 26)
After
y self-inner-product first-slot term1 = 104
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(114, 104)
After
y self-inner-product inner sum = 218
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
parallelogram sum coordinate0 := sum(-2, 3)
After
parallelogram sum 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, S030
Before
parallelogram sum coordinate1 := sum(1, 4)
After
parallelogram sum coordinate1 = 5
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
parallelogram difference coordinate0 := sub(-2, 3)
After
parallelogram difference 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, S032
Before
parallelogram difference coordinate1 := sub(1, 4)
After
parallelogram difference 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, S033
Before
sum self-inner-product:same matrix=[[10, 2], [2, 5]];first vector=[1, 5];second vector=[1, 5]
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, S034
Before
sum self-inner-product matrix term0,0 := mul(10, 1)
After
sum self-inner-product matrix term0,0 = 10
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
sum self-inner-product matrix term0,1 := mul(2, 5)
After
sum self-inner-product matrix term0,1 = 10
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
sum self-inner-product image coordinate0 := sum(10, 10)
After
sum self-inner-product image coordinate0 = 20
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
sum self-inner-product matrix term1,0 := mul(2, 1)
After
sum 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, S038
Before
sum self-inner-product matrix term1,1 := mul(5, 5)
After
sum self-inner-product matrix term1,1 = 25
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
sum self-inner-product image coordinate1 := sum(2, 25)
After
sum self-inner-product image coordinate1 = 27
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
sum self-inner-product first-slot term0 := mul(1, 20)
After
sum self-inner-product first-slot term0 = 20
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
sum self-inner-product first-slot term1 := mul(5, 27)
After
sum self-inner-product first-slot term1 = 135
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
sum self-inner-product inner sum := sum(20, 135)
After
sum self-inner-product inner sum = 155
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
difference self-inner-product:same matrix=[[10, 2], [2, 5]];first vector=[-5, -3];second vector=[-5, -3]
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, S044
Before
difference self-inner-product matrix term0,0 := mul(10, -5)
After
difference self-inner-product matrix term0,0 = -50
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
difference self-inner-product matrix term0,1 := mul(2, -3)
After
difference self-inner-product matrix term0,1 = -6
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
difference self-inner-product image coordinate0 := sum(-50, -6)
After
difference self-inner-product image coordinate0 = -56
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
difference self-inner-product matrix term1,0 := mul(2, -5)
After
difference self-inner-product matrix term1,0 = -10
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
difference self-inner-product matrix term1,1 := mul(5, -3)
After
difference self-inner-product matrix term1,1 = -15
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
difference self-inner-product image coordinate1 := sum(-10, -15)
After
difference self-inner-product image coordinate1 = -25
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
difference self-inner-product first-slot term0 := mul(-5, -56)
After
difference self-inner-product first-slot term0 = 280
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
difference self-inner-product first-slot term1 := mul(-3, -25)
After
difference self-inner-product first-slot term1 = 75
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
difference self-inner-product inner sum := sum(280, 75)
After
difference self-inner-product inner sum = 355
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
parallelogram left sum := sum(155, 355)
After
parallelogram left sum = 510
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
twice x self-inner-product := mul(2, 37)
After
twice x self-inner-product = 74
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
twice y self-inner-product := mul(2, 218)
After
twice y self-inner-product = 436
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
parallelogram right sum := sum(74, 436)
After
parallelogram right sum = 510
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
same inner product;left=510;right=510
After
bilinear expansions have opposite cross terms which cancel;identity=Yes
Conditions: none
Expand q(x+y),q(x−y);their opposite±2xy terms cancel to2q(x)+2q(y).This instance neither constructs an inner product by converse nor certifies Hilbert completeness.
Rule: parallelogram_same_form_identity_not_Hilbert_claim · Uses: G001, S058
S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S058 · premise_and_result_audit — Recompute parallelogram right sum with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Recompute twice y self-inner-product with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Recompute twice x self-inner-product with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute parallelogram left sum with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Recompute difference self-inner-product inner sum with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute difference self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute difference self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Recompute difference self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute difference self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute difference self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute difference self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute difference self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute difference self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Recompute sum self-inner-product inner sum with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute sum self-inner-product first-slot term1 with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute sum self-inner-product first-slot term0 with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute sum self-inner-product image coordinate1 with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute sum self-inner-product matrix term1,1 with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute sum self-inner-product matrix term1,0 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute sum self-inner-product image coordinate0 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute sum self-inner-product matrix term0,1 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute sum self-inner-product matrix term0,0 with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S034 · premise_and_result_audit — Recompute parallelogram difference coordinate1 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute parallelogram difference coordinate0 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute parallelogram sum coordinate1 with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute parallelogram sum coordinate0 with exact rational arithmetic, not rounded decimals.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.([-2, 1], [3, 4], 510, 510, Yes)
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: 0335e82667e39b3cafd44347ea18ec6ed10c7f65eb575f70d2d59a5f40d1e2dbcandidate record SHA-256: 12374c91998c4bf9efc6bfd35d872308ec1b423a894ff483755f30139f5c8b85semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209aequestion-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0Lean: NOT_RUN · human review: NOT_PERFORMED