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R0023-12221 · q-20261013-advance-20261026-r05-covariance-0071
Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[3, 8, -2, -8]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).
G001 — Use the complete list of distinct integer pairs=[[8, -1], [9, 0], [10, 1], [11, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[3, 8, -2, -8]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).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
joint law total mass := sum(1/8, 1/4, 1/8, 1/2)
After
joint law total mass = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001
Before
original distinct joint pairs=[[8, -1], [9, 0], [10, 1], [11, 2]];raw aligned masses=[[1, 8], [1, 4], [1, 8], [1, 2]]
After
normalized aligned masses=[1/8, 1/4, 1/8, 1/2];nonnegative=Yes;total=1
Conditions: none
Use the complete original joint table on one common probability space,preserving order,pairing and zero masses.Covariance requires nonnegative masses with exact total1.
Rule: qualify_whole_joint_law_preserve_alignment · Uses: G001, S001
Before
original atom0 weighted X := mul(1/8, 8)
After
original atom0 weighted X = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
original atom0 weighted Y := mul(1/8, -1)
After
original atom0 weighted Y = -1/8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
original atom0 XY := mul(8, -1)
After
original atom0 XY = -8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
original atom0 weighted XY := mul(1/8, -8)
After
original atom0 weighted XY = -1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
original atom1 weighted X := mul(1/4, 9)
After
original atom1 weighted X = 9/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
original atom1 weighted Y := mul(1/4, 0)
After
original atom1 weighted Y = 0
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
original atom1 XY := mul(9, 0)
After
original atom1 XY = 0
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
original atom1 weighted XY := mul(1/4, 0)
After
original atom1 weighted XY = 0
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
original atom2 weighted X := mul(1/8, 10)
After
original atom2 weighted X = 5/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
original atom2 weighted Y := mul(1/8, 1)
After
original atom2 weighted Y = 1/8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
original atom2 XY := mul(10, 1)
After
original atom2 XY = 10
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
original atom2 weighted XY := mul(1/8, 10)
After
original atom2 weighted XY = 5/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
original atom3 weighted X := mul(1/2, 11)
After
original atom3 weighted X = 11/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
original atom3 weighted Y := mul(1/2, 2)
After
original atom3 weighted Y = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
original atom3 XY := mul(11, 2)
After
original atom3 XY = 22
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
original atom3 weighted XY := mul(1/2, 22)
After
original atom3 weighted XY = 11
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
original E_X := sum(1, 9/4, 5/4, 11/2)
After
original E_X = 10
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
original E_Y := sum(-1/8, 0, 1/8, 1)
After
original E_Y = 1
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
original E_XY := sum(-1, 0, 5/4, 11)
After
original E_XY = 45/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
original product of means := mul(10, 1)
After
original product of means = 10
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
original covariance := sub(45/4, 10)
After
original covariance = 5/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
original:E_X=10;E_Y=1;E_XY=45/4
After
Cov=E_XY-E_X*E_Y=5/4
Conditions: none
All three expectations use the same atoms and masses.Do not mix marginal models or assume independence.
Rule: covariance_same_joint_three_moments · Uses: G001, S023
Before
affine atom0 scaled X := mul(3, 8)
After
affine atom0 scaled X = 24
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
affine atom0 U := sum(24, 8)
After
affine atom0 U = 32
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
affine atom0 scaled Y := mul(-2, -1)
After
affine atom0 scaled Y = 2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
affine atom0 V := sum(2, -8)
After
affine atom0 V = -6
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
affine atom1 scaled X := mul(3, 9)
After
affine atom1 scaled X = 27
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
affine atom1 U := sum(27, 8)
After
affine atom1 U = 35
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
affine atom1 scaled Y := mul(-2, 0)
After
affine atom1 scaled Y = 0
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
affine atom1 V := sum(0, -8)
After
affine atom1 V = -8
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
affine atom2 scaled X := mul(3, 10)
After
affine atom2 scaled X = 30
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
affine atom2 U := sum(30, 8)
After
affine atom2 U = 38
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
affine atom2 scaled Y := mul(-2, 1)
After
affine atom2 scaled Y = -2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
affine atom2 V := sum(-2, -8)
After
affine atom2 V = -10
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
affine atom3 scaled X := mul(3, 11)
After
affine atom3 scaled X = 33
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
affine atom3 U := sum(33, 8)
After
affine atom3 U = 41
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
affine atom3 scaled Y := mul(-2, 2)
After
affine atom3 scaled Y = -4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
affine atom3 V := sum(-4, -8)
After
affine atom3 V = -12
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
affine transformed atom0 weighted X := mul(1/8, 32)
After
affine transformed atom0 weighted X = 4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
affine transformed atom0 weighted Y := mul(1/8, -6)
After
affine transformed atom0 weighted Y = -3/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
affine transformed atom0 XY := mul(32, -6)
After
affine transformed atom0 XY = -192
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
affine transformed atom0 weighted XY := mul(1/8, -192)
After
affine transformed atom0 weighted XY = -24
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
affine transformed atom1 weighted X := mul(1/4, 35)
After
affine transformed atom1 weighted X = 35/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
affine transformed atom1 weighted Y := mul(1/4, -8)
After
affine transformed atom1 weighted Y = -2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
affine transformed atom1 XY := mul(35, -8)
After
affine transformed atom1 XY = -280
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
affine transformed atom1 weighted XY := mul(1/4, -280)
After
affine transformed atom1 weighted XY = -70
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
affine transformed atom2 weighted X := mul(1/8, 38)
After
affine transformed atom2 weighted X = 19/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
affine transformed atom2 weighted Y := mul(1/8, -10)
After
affine transformed atom2 weighted Y = -5/4
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
affine transformed atom2 XY := mul(38, -10)
After
affine transformed atom2 XY = -380
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
affine transformed atom2 weighted XY := mul(1/8, -380)
After
affine transformed atom2 weighted XY = -95/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
affine transformed atom3 weighted X := mul(1/2, 41)
After
affine transformed atom3 weighted X = 41/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
affine transformed atom3 weighted Y := mul(1/2, -12)
After
affine transformed atom3 weighted Y = -6
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
affine transformed atom3 XY := mul(41, -12)
After
affine transformed atom3 XY = -492
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
affine transformed atom3 weighted XY := mul(1/2, -492)
After
affine transformed atom3 weighted XY = -246
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
affine transformed E_X := sum(4, 35/4, 19/4, 41/2)
After
affine transformed E_X = 38
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
affine transformed E_Y := sum(-3/4, -2, -5/4, -6)
After
affine transformed E_Y = -10
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
affine transformed E_XY := sum(-24, -70, -95/2, -246)
After
affine transformed E_XY = -775/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
affine transformed product of means := mul(38, -10)
After
affine transformed product of means = -380
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
affine transformed covariance := sub(-775/2, -380)
After
affine transformed covariance = -15/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
affine transformed:E_X=38;E_Y=-10;E_XY=-775/2
After
Cov=E_XY-E_X*E_Y=-15/2
Conditions: none
All three expectations use the same atoms and masses.Do not mix marginal models or assume independence.
Rule: covariance_same_joint_three_moments · Uses: G001, S061
Before
affine two scale product := mul(3, -2)
After
affine two scale product = -6
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
affine scaled original covariance := mul(-6, 5/4)
After
affine scaled original covariance = -15/2
Conditions: none
Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
U=(3)*X+(8);V=(-2)*Y+(-8);same indexed atoms and weights
After
E_UV-E_U*E_V=(3)*(-2)*(E_XY-E_X*E_Y)=-15/2
Conditions: none
Expand E[UV]=acE[XY]+adE[X]+bcE[Y]+bd and subtract(aE[X]+b)(cE[Y]+d);all shift terms cancel.Even zero scaling may duplicate image points:retain original probability atoms rather than treating images as a new distinct support.
Rule: affine_covariance_shift_cancellation_not_new_joint_support · Uses: G001, S064
S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S064 · premise_and_result_audit — Recompute affine scaled original covariance with exact rational arithmetic, not rounded decimals.S063 · premise_and_result_audit — Recompute affine two scale product with exact rational arithmetic, not rounded decimals.S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S061 · premise_and_result_audit — Recompute affine transformed covariance with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Recompute affine transformed product of means with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute affine transformed E_XY with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute affine transformed E_Y with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Recompute affine transformed E_X with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Recompute affine transformed atom3 weighted XY with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute affine transformed atom3 XY with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Recompute affine transformed atom3 weighted Y with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute affine transformed atom3 weighted X with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute affine transformed atom2 weighted XY with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Recompute affine transformed atom2 XY with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute affine transformed atom2 weighted Y with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute affine transformed atom2 weighted X with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute affine transformed atom1 weighted XY with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute affine transformed atom1 XY with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute affine transformed atom1 weighted Y with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Recompute affine transformed atom1 weighted X with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Recompute affine transformed atom0 weighted XY with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute affine transformed atom0 XY with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute affine transformed atom0 weighted Y with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute affine transformed atom0 weighted X with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute affine atom3 V with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Recompute affine atom3 scaled Y with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute affine atom3 U with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute affine atom3 scaled X with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute affine atom2 V with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute affine atom2 scaled Y with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute affine atom2 U with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute affine atom2 scaled X with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute affine atom1 V with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute affine atom1 scaled Y with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute affine atom1 U with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute affine atom1 scaled X with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute affine atom0 V with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute affine atom0 scaled Y with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute affine atom0 U with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute affine atom0 scaled X with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Recompute original covariance with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute original product of means with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute original E_XY with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute original E_Y with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute original E_X with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute original atom3 weighted XY with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute original atom3 XY with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute original atom3 weighted Y with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute original atom3 weighted X with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute original atom2 weighted XY with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute original atom2 XY with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute original atom2 weighted Y with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute original atom2 weighted X with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute original atom1 weighted XY with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute original atom1 XY with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute original atom1 weighted Y with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute original atom1 weighted X with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute original atom0 weighted XY with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute original atom0 XY with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute original atom0 weighted Y with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute original atom0 weighted X with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Recompute joint law total mass with exact rational arithmetic, not rounded decimals.(3, 8, -2, -8, 5/4, -15/2, Cov(aX+b,cY+d)=ac Cov(X,Y))
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: cfab7fe69011e374f633a7362db3d62c10ff17c91089623fa64df089c45b603acandidate record SHA-256: c9187177a8f35fd11d4e65f53412f305817de6dc75546bd49b409a88a0acf419semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ffquestion-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18Lean: NOT_RUN · human review: NOT_PERFORMED